The warning came after weeks of alerts from U.S. and U.N. officials that conditions were deteriorating further in Gaza, particularly in the territory’s north, amid increasing Israeli restrictions on the delivery of international aid.

Mr. Miller said the level of humanitarian aid into Gaza was the lowest it has been at any time since the Hamas-led attacks on Israel on Oct. 7 last year. He did not specify the possible consequences Israel faced if it did not allow more aid into Gaza, although a copy of the warning letter posted online by a reporter for Axios clearly raised the possibility that the United States could suspend military aid to Israel. U.S. law bars providing military aid to any country found to be blocking the delivery of U.S.-provided humanitarian aid.

Israel has intensified its operations in Gaza again as it seeks to destroy Hamas, even as it has launched a ground invasion of Lebanon to battle Hezbollah, which is, like Hamas, backed by Israel’s arch-nemesis Iran. The Israeli government has told the Biden administration that it will avoid striking Iran’s nuclear enrichment and oil production sites when it responds to Iran’s Oct. 1 missile attack on Israel, two officials said, a move that may reduce the immediate likelihood of an all-out war between the two adversaries.

However, the acting leader of Hezbollah, Sheikh Naim Qassem, said it would seek to expand its conflict with Israel by launching strikes deeper inside Israeli territory. In a televised address on Tuesday, he signaled that Hezbollah would not agree to a truce with Israel unless a cease-fire was reached between Israel and Hamas in Gaza. Israeli attacks over the past several weeks have killed or injured many of Hezbollah’s top leaders and fighters, though the group has continued to fire rockets at Israeli cities.

Here are other developments:

THAAD crew arrives: A team of U.S. military personnel has reached Israel before the arrival of the Terminal High Altitude Area Defense system, which aims to thwart attacks by Iran, the Pentagon said in a statement. It did not say when the missile defense system would be operational. The Pentagon said over the weekend that it was deploying the THAAD battery and about 100 American troops to Israel.

Northern Lebanon: After an Israeli airstrike on Monday killed at least 21 people in the Lebanese village of Aitou, according to health officials, the Israeli military said that it had “struck a target belonging to the Hezbollah terrorist organization.” The military added that it was reviewing reports that civilians were killed in the strike. Aitou is in a largely Christian region of northern Lebanon that is not known to have been targeted before by Israel as it steps up its assault on Hezbollah.

Health system attacked: Israeli airstrikes overnight damaged a hospital in Baalbek, in eastern Lebanon, and put it out of service, Bachir Khodr, the regional governor, said Tuesday. Lebanon’s health minister, Firass Abiad, said that 13 hospitals in the country had now been completely or partially shut down, and that Israeli attacks had killed over 150 paramedics and health workers since hostilities with Hezbollah began last year

Radwan soldiers captured: The Israeli military said that it had captured three members of Hezbollah’s elite Radwan unit in southern Lebanon and that the three had been taken to an interrogation facility inside Israel. There was no immediate comment from Hezbollah. The Radwan unit, better trained and equipped than Hezbollah’s regular forces, has taken the lead in much of the fighting with Israeli troops.

Iranian general: Brig. Gen. Esmail Ghaani, the commander of Iran’s elite Quds Force, appeared in video footage broadcast by Iranian state news media and photographs published by independent international news media, ending days of speculation about his whereabouts. The Quds Force is the branch of the Islamic Revolutionary Guards Corps that conducts external operations, including liaisons with Hezbollah, Hamas and other militant groups that Iran supports.

Patrick Kingsley contributed reporting from Jerusalem and Ronen Bergman from Tel Aviv.

U.S. and United Nations officials have warned in recent weeks that conditions are deteriorating further in Gaza, particularly in the territory’s north, as Israel has placed increasing restrictions on the delivery of international aid.

“What we have seen over the past few months is that the level of humanitarian assistance has not been sustained,” Mr. Miller said at a daily news briefing. “In fact, it has fallen by over 50 percent from where it was at its peak.”

He added that Mr. Blinken and Mr. Austin “thought it was appropriate to make clear to the government of Israel that there are changes that they need to make” to increase aid deliveries “from the very, very low levels” of the current moment. He said the level of humanitarian aid into Gaza in September was the lowest it has been at any time since the Hamas-led Oct. 7 attacks in Israel.

Mr. Miller would not specify the possible consequences if Israel did not comply, although a copy of the letter posted online by Barak Ravid, a reporter for Axios, clearly raised the possibility of suspending military aid. U.S. law bars providing military aid to any country found to be blocking the delivery of U.S.-provided humanitarian aid.

“We are now writing to underscore the U.S. government’s deep concern over the deteriorating humanitarian situation in Gaza, and seek urgent and sustained actions by your government this month to reverse this trajectory,” the copy of the letter states.

The 30-day deadline set by the letter would fall after the U.S. presidential election on Nov. 5, potentially making it easier for President Biden to take stronger action against Israel than he has so far been willing.

Mr. Miller said that the 30-day period was set to provide Israel with “an appropriate period of time to implement” changes to its aid delivery, rather than demanding that “this has to happen overnight.”

“It’s just a plain reading of U.S. law,” Mr. Miller said. “We are required to conduct assessments and find that recipients of U.S. military assistance do not arbitrarily deny or impede the provisioning of us humanitarian assistance. That’s just the law, and we of course will follow the law.”

He added that the letter was intended to be a private diplomatic correspondence, but that the U.S. was confirming its existence after it was leaked to the media.

Mr. Blinken sent a similar letter to Mr. Gallant in April, which Mr. Miller credited with prompting Israel to increase its aid deliveries into Gaza. After that letter, he said, as many as 300 to 400 aid trucks were entering Gaza on some days — a level U.S. officials consider adequate. But, he added, Mr. Blinken also made clear at that time “that the increase couldn’t be a one-off, that it needed to be sustained.”

Eric Schmitt contributed reporting.

Her neighbors have since fled, but Ms. Soliman, who lives with several other women and a 7-year-old, said she cannot join them because her mother uses a wheelchair and is unable to travel.

“I simply cannot take her out,” Ms. Soliman said by telephone from her apartment in Jabaliya, which has been shattered by Israeli airstrikes over the past year. “I cannot leave her either,” she added, “So we all stayed.”

Her dilemma mirrors that of many civilians in northern Gaza, a part of the enclave that is the subject of mounting international concern and rising casualties. It is not clear how many people have obeyed military orders to evacuate north Gaza but around 400,000 people remain, according to U.N. estimates. Some, including the sick and disabled, cannot leave. Some argue that they might face greater danger on the road or at their destination.

The International Committee of the Red Cross said on Tuesday that thousands of families in the area have been displaced and that those who remain are facing “unimaginable fear” as well as the loss of loved ones.

UNRWA, the U.N. agency that aids Palestinian refugees, said on Monday that Israeli tank shells had hit a food distribution center in Jabaliya while people were collecting food. It cited reports that 10 people had been killed. The Israeli military said it was investigating the reports, which could not be verified independently.

Crews with the Palestinian civil defense, an emergency service in Gaza, retrieved 12 bodies and evacuated three wounded people from sites across Jabaliya on Tuesday, according to Dr. Mohammed Al Moghayer, a spokesman for the organization. The organization said at least 69 people had been killed in northern Gaza since Oct. 5, when the Israeli operation began. The claims could not be independently verified. Israel’s military said it was looking into the report.

The area has become dangerous for both civilians and rescue workers. The Gaza ministry of health said that a doctor, Ahmed Talab al-Najjar, was killed on Tuesday while trying to evacuate injured people in the al-Falluja neighborhood of Jabaliya.

In response to Israel’s evacuation order, the Palestinian Red Crescent pulled its staff out of Jabaliya and focused on calls to evacuate the wounded and retrieve bodies. Calls to provide rescue services sometimes go unheeded because of damage to roads and because the Israeli military denies access, Nebal Farsakh, a spokeswoman for the organization, said on Tuesday. Israel’s military said it was looking into the report.

The U.N. human rights office said on Monday that, despite the evacuation order, Israeli forces appeared to be sealing off north Gaza by erecting sand barriers at a key crossing point. The military had also reportedly “opened fire, killing some Palestinians trying to evacuate” farther south, it said.

Israel’s military said it was investigating the report. On Tuesday, Israel’s military and the government organization that coordinates the country’s policy in Gaza and the West Bank, COGAT, listed a series of measures it had taken to provide medical support and food aid for civilians in northern Gaza.

Amid the chaos, Ms. Soliman said that Israeli soldiers had telephoned three times with the message to evacuate and each time she told them she could not. She also said that the Red Cross had telephoned to say that it had informed Israel’s military of her presence and her difficulties.

“We have some canned food and water stored at home,” she said, adding that they are using as little water as possible because they can’t get anymore.

The officials, who spoke on condition of anonymity to discuss sensitive diplomacy, said that Israel had agreed to focus its next attack on military targets in Iran, instead of those related to Iran’s oil industry or its uranium enrichment efforts. The pledge was first reported by the Washington Post.

While no final decision is believed to have been made, an Israeli retaliation could still be large in scale, possibly prompting Iran to continue the cycle of attacks. And the officials said that Israel’s assurances related only to its next attack — meaning that it could still pursue more ambitious targets in future rounds of fighting with Iran.

For decades, Iran has sought Israel’s destruction while Israel has pushed to collapse the Iranian regime, leading to a decades-long shadow war in which each side has secretly attacked the other’s interests and supported the other’s enemies.

That covert war has broken out into the open in recent months, partly because of Israel’s war with Hamas, an ally and proxy of Iran. Hamas unsuccessfully tried to persuade Iran to participate in the attack on Israel last October that prompted Israel to invade Gaza, according to documents obtained by The Times, but Iran has supported Hamas with funds and diplomatic support.

Iran fired a huge barrage of drones and ballistic missiles at Israel in April after Israel killed several Iranian commanders. Israel responded by striking an Iranian radar station and later killed a Hamas leader, Ismail Haniyeh, as he visited Tehran. This month, Iran fired another huge round of missiles at Israel, partly in response to Mr. Haniyeh’s assassination — setting the Middle East further on edge as it awaits Israel’s next move.

U.S. officials believe that if Israel goes after Iran’s most sensitive sites, the result could be an uncontrolled escalation. President Biden and Prime Minister Benjamin Netanyahu of Israel spoke last week for the first time in months as relations between the two allies have sharply deteriorated, with neither side publicly mentioning Israel’s plans to respond to Iran.

In a statement on Tuesday, Mr. Netanyahu’s office said: “We listen to the opinions of the United States, but we will make our final decisions based on our national interests.” The U.S. Embassy in Israel declined to comment.

Even if it avoids Iran’s nuclear enrichment and oil sites, Israel could still hit a wide array of military targets. They include missile and drone launchers, missile and drone storage sites, missile and drone factories, as well as military bases and major government buildings, according to two Israeli officials briefed on the planning process, who spoke on condition of anonymity to discuss military matters. These two officials said that Israel could also strike nuclear research laboratories, even if it avoids Iran’s subterranean nuclear enrichment sites.

Mr. Netanyahu met with security chiefs at an intelligence base on Sunday to discuss the plan and his government has yet to agree on a specific approach, the two Israeli officials said.

The officials added that some Israeli leaders want to reduce tensions with Iran to reach a truce in Lebanon, where the Israeli military has launched an air and ground campaign targeting Hezbollah, the Iranian-backed Lebanese militia. Other Israeli leaders feel they have a rare opportunity to destroy Iran’s nuclear facilities at some point in the near future, and believe that fueling a bigger confrontation with Iran would help create a pretext for such a strike during future rounds of conflict, the officials said.

Gabby Sobelman and Myra Noveck contributed reporting.

Mr. Wadi said this was the seventh strike on the hospital his family had witnessed since setting up a tent outside the facility. This time, instead of awakening in a daze to the sight of smoke rising from one spot in the camp, the heat of flames was everywhere, he said. He saw bodies “scorched and black, like giant lumps of coal.”

The Wadi family is one of scores of families that have set up camp in the parking lot of the compound, hoping that international laws prohibiting attacks on hospitals made the area a safe place to shelter. Instead, these families say, they have survived repeated strikes on the hospital. The latest attack, shortly after 1 a.m. on Monday, triggered a fire that set the camp ablaze.

The Israeli military said in a statement posted to social media that it had been targeting a Hamas command center located near the hospital. The fire that erupted afterward was likely caused by secondary explosions, it said.

Survivors interviewed amid the smoldering remains of the camp told The New York Times that the fast-moving fire had been fueled by the explosions of families’ cooking gas canisters and flames that fed off their plastic tents.

“The most difficult scene you can experience is seeing your neighbors burning alive and not being able to do anything to rescue them,” said Abed Musleh, a 25-year-old who fled northern Gaza and was sheltering in a tent in the parking lot with his wife, two children, and his four sisters. He estimated the fire burned at least 30 tents. Residents scrambled to find any buckets not burned in the blaze to try and help rescuers put out the fire.

The Palestinian health authority said four people died and over a dozen were injured, but that the death toll would likely rise. Later Monday, Doctors Without Borders, which has medics operating in Gaza, said five people were killed and dozens were wounded, some with severe burn injuries.

Despite the repeated strikes, Mr. Musleh had no plans to leave. He cannot find anywhere else to go, he said, and he still couldn’t imagine that any other place would be safer than a hospital.

Israel has come under repeated criticism for hitting civilian infrastructure, including hospitals, during the yearlong war in Gaza. A U.N. report last week accused Israel of a deliberate policy to destroy the health care system. The Israeli military has said it acted on information that Hamas was operating from the hospital compound, and it has repeatedly said that it tries to avoid civilian casualties.

But for Gaza’s two million people, an increasingly common refrain is that nowhere is safe — and that, with more than 90 percent of the population displaced, there are few places left to go.

Mohammed Ramadan, whose family of 10 survived but lost their tent, said he felt trapped by impossible options: “There are no safe places, and no places left to shelter in.”

The THAAD adds another layer of protection to the several types of air defense systems that Israel already uses to shoot down missiles. About 100 American troops will go to Israel to operate the THAAD system, according to the Pentagon, putting U.S. service members closer to the heart of a widening Middle East war.

“It’s a political message of the United States to Israel that, ‘We are with you,’” said Yehoshua Kalisky, a military technology expert at the Institute for National Security Studies in Tel Aviv. “And to enemies, it’s: ‘Don’t.’”

Here is a look at the THAAD system and what it can do.

What is THAAD?

The Terminal High Altitude Area Defense system is a mobile surface-to-air interceptor designed to shoot down incoming ballistic missiles. It is categorized as a short-, medium- and intermediate-range interceptor that can strike incoming targets both within the Earth’s atmosphere and above it.

Each THAAD battery is made up of five parts: 48 interceptor missiles, six truck-mounted launchers, a radar, a command-and-control platform and 95 soldiers, according to the Congressional Research Service. There is no warhead on the missile, which destroys its targets by the force of its impact.

There are only nine active THAAD batteries in the world, according to its manufacturer, Lockheed Martin. In 2019, the latest data available, seven of them were assigned to the U.S. Army, including five at bases in Texas and one each Guam and South Korea. The battery that the Biden administration has ordered to Israel presumably would be among those seven.

Two others are being fielded in the United Arab Emirates.

The Pentagon announced almost a year ago that it was sending a THAAD battery to the Middle East to help protect Israel, but did not specify where it would be.

How will it be used?

Because it can reach above the atmosphere, the THAAD should be able to intercept ballistic missiles launched from Iran and Yemen, said Fabian Hinz, a missiles and Middle East expert at the International Institute for Strategic Studies in London. It can also shoot down shorter-range missiles launched by Hezbollah, in neighboring Lebanon.

Iran and its allies have also fired guided cruise missiles and drones, which operate at much lower altitudes and speeds than ballistic missiles.

Mr. Kalisky said the THAAD would be particularly useful in intercepting debris from other missiles that have been downed before it falls to the ground, where it can inflict casualties and damage infrastructure. (He also said the THAAD has an exceptional radar that can detect incoming missiles from farther distances.)

But Mr. Hinz said they would likely serve as another layer of urgently needed air defenses, given that some Iranian missiles evaded Israeli interceptors during a barrage this month. Iran lies more than 500 miles from Israel.

“We have seen that the Iranian strategy is to fire large volleys in order to overwhelm Israeli defenses,” Mr. Hinz said. “If you have additional interception capability, that is quite useful.”

How is it different from Israel’s other air defense systems?

It’s largely a matter of range, meaning how far the missile can fly. The THAAD has a range of about 125 miles. Its launchers and command centers can be moved to different sites.

Mr. Kalisky compared the THAAD to one of Israel’s main defense systems, the David’s Sling, a stationary weapon at a fixed location that can shoot down short- and medium-range ballistic and cruise missiles. It has a range of about 185 miles and is also a “hit-to-kill” weapon that downs its targets by flying into it. The David’s Sling is jointly produced by Raytheon and Israeli weapons producer Rafael.

Then there is Israel’s Arrow series, which is produced by Israel Aerospace Industries and Boeing.

The Arrow 2 can intercept targets high in the atmosphere, with an altitude of about 30 miles and a range of about 60 miles. It has a fragmentation warhead packed with explosives that can blow up near incoming missiles even if it does not directly hit its targets.

The Arrow 3, another hit-to-kill weapon, can go beyond the atmosphere with a range of up to 1,500 miles. It is one of Israel’s most advanced defenses and was used to counter the Iranian strikes on Oct. 1. Both the Arrow 2 and the Arrow 3 are ground-based mobile launchers.

The Iron Dome system is perhaps the best-known of Israel’s air defenses, largely because it is used more than the others. Its short-range interceptors — just 6 inches wide and 10 feet long — rely on miniature sensors and computerized guidance to zero in on short-range rockets. It is produced by Rafael, the Israeli defense contractor.