Democratic anxiety over President Biden’s fitness to run for re-election broke out into the open on Tuesday in a spike of panic, as the first sitting member of Congress called on Mr. Biden to withdraw and a slew of other prominent officials who have backed the president vented their concerns.

One Democratic senator openly asked for assurances from the White House about Mr. Biden’s “condition” — “That this was a real anomaly, and not just the way he is these days,” said Senator Sheldon Whitehouse of Rhode Island, who told a local television station he had been “horrified” by Mr. Biden’s performance.

Another Democratic senator, Peter Welch of Vermont, scolded the Biden campaign for “a dismissive attitude towards people who are raising questions for discussion.”

And later on Tuesday, Representative Lloyd Doggett of Texas, a veteran progressive lawmaker, issued a statement saying that Mr. Biden’s debate performance had disqualified him from running again.

“I had hoped that the debate would provide some momentum to change that. It did not,” Mr. Doggett said. “Instead of reassuring voters, the president failed to effectively defend his many accomplishments and expose Trump’s many lies.”

Mr. Biden’s fumbling, faltering debate performance on Thursday set off a quiet panic among party officials and activists over whether he should be replaced on the ticket roughly four months before Election Day. But it also unleashed fresh anxiety among lawmakers, strategists and operatives about the effect of his stumbles on his party’s ability to win the critical races that will decide control of the House and Senate.

“He clearly has to understand,” Representative Mike Quigley, Democrat of Illinois, said on CNN on Tuesday, “that his decision not only impacts who is going to serve in the White House the next four years, but who is going to serve in the Senate, who is going to serve in the House, and it’s going to have implications for decades to come.”

Democrats in congressional races across the country have long understood that they would need to outperform Mr. Biden in order to win their seats. In that sense, several Democratic operatives working on congressional races said, Mr. Biden’s performance would do little to affect the strategy in down-ballot elections. But the fresh injection of doubt over their party leader’s fitness to serve has confronted them with a new series of difficult calculations about their own races.

For now, leading Democrats are expressing strong backing for Mr. Biden. Senator Chuck Schumer of New York, the majority leader, said during an appearance in Syracuse on Tuesday that, “yes,” he believed the president was fit to serve.

“I’m with Joe Biden,” he said.

But some are also conceding that the party has grave concerns about the situation. Representative Nancy Pelosi, Democrat of California and the former speaker, said it was “legitimate” to ask whether Mr. Biden’s bad night at the debate was a one-time flub or “a condition,” adding that former President Donald J. Trump should have to answer the same question about the falsehoods he uttered.

And she conceded in an interview on MSNBC that she was hearing “mixed” feedback from Democratic donors on whether the president was up for running for another term in office.

In a pair of statements from Democrats’ House and Senate campaign arms, spokesmen each stressed that congressional races are “candidate vs. candidate battles,” as David Bergstein, the Senate campaign communications director said.

“House races have always been about the strength of our candidates, combined with the fact that Democrats deliver when in charge while extreme Republicans sow chaos,” Viet Shelton, the spokesman from House Democrats’ campaign arm, said. “It’s why recent polling has been showing Democrats outrunning their Republican opponents across the battleground. That hasn’t changed after the debate.”