Moving to head off potential backlash, Mr. McConnell told reporters pressing him to rebuke Mr. Trump’s rhetoric that he did not have “any advice” for Mr. Trump. Earlier in the day before congratulating Mr. Biden, he had used his speech on the Senate floor to lavish praise on the president’s record on foreign and domestic policy.

The blowback was immediate from the party’s outspoken right flank anyway and foreshadowed the return to an old dynamic briefly abated during the Trump years in which Mr. McConnell was a favorite punching bag for conservatives. Mark Levin, the talk radio host and strident supporter of Mr. Trump, declared that Mr. McConnell had been “AWOL” from “challenging the lawless acts of the Biden campaign and Democrats.”

“Trump helped you secure your seat, as he did so many Senate and House seats, and you couldn’t even wait until January 6th,” Mr. Levin wrote on Twitter. “You’ve been the GOP ‘leader’ in the Senate for far too long. It’s time for some fresh thinking and new blood.”

Nor did Mr. McConnell earn much love from the few voices of Republican dissent that have raised alarms in recent days that Mr. Trump’s defiance of democratic norms — and the acquiescence of much of his party — would do lasting damage both to the G.O.P. and to the country.

One of them, Representative Paul Mitchell of Michigan, who is retiring, went as far as to quit the party on Monday in protest. Another, Senator Mitt Romney of Utah, the party’s 2012 presidential nominee, said on Tuesday that merely recognizing Mr. Biden’s victory was not enough for his party.

“How many Republicans will say that what the president is saying is simply wrong and dangerous?” Mr. Romney said on CNN. “We need to have people who are strong Trump supporters say that as well, or you are going to continue to have this country divided, which is pretty dangerous.”

Representative Tom Cole of Oklahoma, a senior Republican committee leader and former head of the party’s campaign arm, argued that fears like Mr. Romney’s were somewhat overwrought, but reflected a general loss of trust by many Americans in the electoral process.