The fallout included at least one resignation, as a key aide to the head of a prominent right-wing think tank stepped down after backing his boss’s vigorous defense of Mr. Carlson.

Kevin Roberts, the president of the Heritage Foundation think tank, had announced late last week that the aide, Ryan Neuhaus, was simply leaving his chief of staff position for another role. But on Monday, a spokesman for the Heritage Foundation, said Mr. Neuhaus had resigned. The resignation was reported earlier by The Hill.

Ben Shapiro, the conservative podcast host, also condemned Mr. Carlson on Monday as “the most virulent superspreader of vile ideas in America,” criticizing him for failing to push back on Mr. Fuentes during the interview and for allowing him instead to spread his ideas unchallenged on a huge platform.

“There’s already the Democratic Party that is anti-Israel, and is OK with antisemitism,” Senator Rick Scott, Republican of Florida, said in an interview. “We’ve got to be very clear we don’t support antisemitism and we do support Israel.”

The uproar over Mr. Carlson’s interview has created a dilemma for many Republicans in Congress. Many have routinely derided “cancel culture” among progressives and accused the left of intolerance. They have also rejected the idea that conservatives should cast out figures within their own ranks who make indefensible statements.

But others, including Senator Ted Cruz, Republican of Texas, have argued that Republicans must rid their movement of such viewpoints. Mr. Cruz has positioned himself as one of the party’s loudest voices denouncing antisemitism and appeared especially eager for a hand-to-hand fight with Mr. Carlson.

Mr. Cruz’s personal and ideological grievances with Mr. Carlson are not new. He was a guest on Mr. Carlson’s show in June, when Mr. Carlson tried to embarrass him by putting him on the spot about his knowledge of Iran. Mr. Carlson said that Mr. Cruz “doesn’t know anything about Iran” while noting that the senator had called for regime change there.

On Monday, Mr. Cruz declined to comment about the resignation of Mr. Neuhaus, the Heritage Foundation aide, or to expound further on his views about Mr. Carlson.

“I am a big fan of the Heritage Foundation,” he said. “I spoke at length on the topic of Tucker Carlson and Nick Fuentes on Thursday night.”

On Friday, Mr. Roberts thrust the right-wing think tank he leads, which created the “Project 2025” plan that has guided much of President Trump’s agenda, into the middle of the controversy.

Many Republican lawmakers, who have made backing Israel a litmus test and taken Democrats to task, accusing them of being insufficiently supportive of the Jewish state, quickly took a different tack.

“I’m in the ‘Hitler sucks’ wing of the Republican Party,” Senator Lindsey Graham, Republican of South Carolina, said over the weekend at the R.J.C. event. “Here’s what I do know: You can sit in a basement with weird people and say weird things. It’s a free country.”

But, Mr. Graham added: “I want the world to know antisemitism, anti-Israel rhetoric, anti-Israel thought is not the road to being elected as a Republican. You will lose.”

Representative Byron Donalds, Republican of Florida, told Breitbart News that antisemitism was: “a cancer, it will be a cancer to Republicans as it is a cancer to America. We have to exhibit moral clarity.”

The statements reflected the political realities the G.O.P. faces. A Gallup poll from February showed that 83 percent of Republicans had a favorable view of Israel. Even with negative views of Israel rising, a recent report from the Brookings Institution said 63 percent of Republicans still held a positive opinion.

In an interview, Matt Brooks, the executive director of the Republican Jewish Coalition, said it was a mistake for Republicans to debate whether it was appropriate for Mr. Carlson to be “platforming” Mr. Fuentes.

“That’s a term of the left,” he said. “Our issue isn’t so much that Tucker had Nick Fuentes on for an interview. Our issue is that he failed to meet the moment and ask him tough questions about why he admires Adolf Hitler, why he’s a Holocaust denier and hates Jews, why he is pro-Putin and pro-Stalin.”

Mr. Brooks said it was unfortunate that the “Hitler is cool” wing of his party was gaining attention, but he noted that it was far from a new phenomenon.

Antisemitism in the Republican Party can be traced all the way back to the John Birch Society, a semi-secret society that espoused antisemitic views. Mr. Carlson remains a formidable force in the Republican Party, and some of the biggest critics in the days after the Fuentes interview were those who had warned of his dangerous influence for years.

Senator Mitch McConnell, Republican of Kentucky, has long blamed Mr. Carlson for the cratering of political support for Ukraine within the Republican Party. Mr. Carlson in the past described Ukrainian President Volodymyr Zelensky, who is Jewish, as “sweaty and rat-like”; “shifty”; “dead-eyed”; a “persecutor of Christians”; and a “friend of BlackRock,” the investment company.

Over the weekend, Mr. McConnell criticized Mr. Roberts for defending Mr. Carlson.

“Last I checked, ‘conservatives should feel no obligation’ to carry water for antisemites and apologists for America-hating autocrats,” he wrote on social media, making a reference to Mr. Roberts’ defense of Mr. Carlson. “But maybe I just don’t know what time it is.”