Six weeks before Election Day, the Republican Party in North Carolina is in an extraordinary standoff with its own nominee for governor.

After a CNN report last week linked the candidate, Lt. Gov. Mark Robinson, to offensive comments on a porn website, donations to his campaign have dried up. Most of his staff has resigned. Incensed Republican officials, donors and lawmakers want him to drop out, but Mr. Robinson has refused to quit.

The question now, party leaders say, is whether he will sink other Republican state and local candidates with him.

The deadline to replace Mr. Robinson on the ballot passed last Thursday, hours after the report that he had written years ago on the porn site that he was a “black NAZI,” that slavery was not bad, and that, as a teenager, he had gone “peeping” on women in public gym showers.

If anyone manages to force him to leave the race, his name will remain on the ballot. In one scenario, the state party would need to find a replacement candidate and explain to the electorate that a vote for Mr. Robinson is technically a vote for the replacement. It remains unclear who would want to run under the tarnished Robinson banner.

If Mr. Robinson remains on the ticket, though, he will likely continue to face a barrage of negative press that could hurt down-ballot candidates and, possibly, former President Donald J. Trump, who believes he must win North Carolina to win the election.

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