President Biden hoped to build fresh momentum for his re-election bid by agreeing to debate nearly two months before he is to be formally nominated. Instead, his halting and disjointed performance on Thursday night prompted a wave of panic among Democrats and reopened discussion of whether he should be the nominee at all.

Over the course of 90 minutes, a raspy-voiced Mr. Biden struggled to deliver his lines and counter a sharp though deeply dishonest former President Donald J. Trump, raising doubts about the incumbent president’s ability to wage a vigorous and competitive campaign four months before the election. Rather than dispel concerns about his age, Mr. Biden, 81, made it the central issue.

Democrats who have defended the president for months against his doubters — including members of his own administration — traded frenzied phone calls and text messages within minutes of the start of the debate as it became clear that Mr. Biden was not at his sharpest. Practically in despair, some took to social media to express shock, while others privately discussed among themselves whether it was too late to persuade the president to step aside in favor of a younger candidate.

“Biden is about to face a crescendo of calls to step aside,” said a veteran Democratic strategist who has staunchly backed Mr. Biden publicly. “Joe had a deep well of affection among Democrats. It has run dry.”

“Parties exist to win,” this Democrat continued. “The man on the stage with Trump cannot win. The fear of Trump stifled criticism of Biden. Now that same fear is going to fuel calls for him to step down.”

A group of House Democrats said they were watching the debate together, and one, who spoke on the condition of anonymity, acknowledged that it was a “disaster” for Mr. Biden. The person said the group was discussing the need for a new presidential nominee.

Mark Buell, a prominent donor for Mr. Biden and the Democratic Party, said after the debate that the president had to strongly consider whether he is the best person to be the nominee. “Do we have time to put somebody else in there?” Mr. Buell said.

He added that he was not yet calling for Mr. Biden to withdraw but that “Democratic leadership has a responsibility to go to the White House and clearly show what America’s thinking, because democracy is at stake here and we’re all nervous.”

Mr. Biden’s goal in accepting a general election debate earlier than ever held in presidential history was to recalibrate the contest as a choice between himself and a felon who tried to overturn an election and would in his view destroy American democracy if given the power of the presidency again. Mr. Biden left the CNN studio in Atlanta instead facing a referendum on himself and his capacity that will reverberate for days if not longer.

Mr. Trump, 78, appeared to coast through the debate with little trouble, rattling off one falsehood after another without being effectively challenged. He appeared confident while avoiding the excessively overbearing demeanor that had damaged him during his first debate with Mr. Biden in 2020, seemingly content to let his opponent stew in his own difficulties.

While Mr. Trump at times rambled and offered statements that were convoluted, hard to follow and flatly untrue, he did so with energy and volume that covered up his misstatements, managing to stay on offense even on issues of vulnerability for him like the Jan. 6, 2021, attack and abortion.

Mr. Biden appeared on defense much of the time and either did not use lines teed up for him by his campaign’s predebate advertising or mumbled them in passing in such a way that they barely registered.

Speaking with reporters afterward, Mr. Biden indicated that he had been battling a cold. “I have a sore throat,” he said. But he expressed satisfaction with his showing. “I think we did well.” Asked about Democrats’ concerns about his showing and calls for him to consider leaving the race, he said: “No. It’s hard to debate a liar.”

Mr. Biden’s advisers have long dismissed any speculation about him dropping out, rejecting it as unjustified nervousness even as he has trailed Mr. Trump in battleground states needed for victory this fall. Biden aides and allies have repeatedly challenged the polls and pointed out that predictions of Democratic defeats in recent elections have been overblown. One reason they cited for an early debate was to make clear to the public that these are the two choices, and no one else will be nominated.

“Folks, the facts are if Joe Biden was going to step aside, he would have done so a long time ago,” said Symone Sanders, a former aide to Vice President Kamala Harris. “That’s not my opinion, that’s literally the facts. So no, he won’t be stepping aside tomorrow morning. He’s the nominee, and a number of Dems I suspect will be out defending him over the next few days.”

The campaign quickly dispatched Ms. Harris to defend the president on CNN after the debate, although even she acknowledged that “it was a slow start, that’s obvious to everyone.” Mr. Biden had shown he could handle the job because of his many accomplishments for Americans, she said, and “the Joe Biden I work with every day is someone who has performed in a way that has been about bringing people into the Oval Office.”

Gov. Gavin Newsom of California, one of those mentioned as a possible replacement for Mr. Biden other than Ms. Harris, brushed off talk about switching candidates. “I would never turn my back on President Biden’s record,” he told reporters, serving as an official surrogate for the campaign in the spin room after the debate. “I would never turn my back on President Biden, and I don’t know a Democrat in my party who would do so, especially after tonight.”

But that did not stop the speculation. “Guys, the Dems should nominate someone else — before it’s too late,” Andrew Yang, who ran against Mr. Biden for the Democratic nomination in 2020, wrote on social media before the debate had ended, adding a hashtag #swapJoeout.

Former Senator Claire McCaskill, Democrat of Missouri, called it “a crisis,” saying that her phone was “blowing up” with senators, operatives, donors and other distraught Democrats doing “more than hand-wringing” about what happens next.

“Joe Biden had one thing he had to do tonight, and he didn’t do it,” she said on MSNBC. “He had one thing he had to accomplish, and that was reassure America that he was up to the job at his age, and he failed at that tonight.”

That judgment extended beyond the political class. Mr. Biden’s perceived odds of winning the nomination plummeted within hours on PredictIt.org, a betting site that takes wagers on political events. His chances of being the party’s candidate fell to 60 cents, down 26 cents, meaning that bettors essentially thought there was only a 60 percent chance of him being nominated even though he swept the primaries, has no internal opponent and controls the party apparatus.

No incumbent president has dropped out of the race so late in the campaign cycle, and there was little consensus about what would happen if he were to. On Thursday night, Democrats were imagining scenarios in which party elders like Senator Chuck Schumer of New York, former Speaker Nancy Pelosi of California and Representative James E. Clyburn of South Carolina were to intervene with Mr. Biden.

There was no indication that any of them would agree to do so. Other Democrats said they feared it was too late, noting that Mr. Biden is a proud, stubborn man who has long insisted he is the best equipped to defeat Mr. Trump and would not listen to anyone other than perhaps his wife, Jill Biden, who has strongly supported another run. Democrats have long fretted that there is no obvious successor, uncertain that Ms. Harris, Mr. Newsom or any other party figure could rise to the challenge.

Incumbent presidents often stumble in their first debate of a general-election season, either because they are rusty or overconfident, but in many cases make up for it with stronger subsequent performances. Mr. Biden’s troubles particularly brought to mind Ronald Reagan’s first debate in 1984, when he appeared old and out of it; he salvaged his campaign at his next debate with a well-timed joke about not exploiting his opponent’s youth and inexperience.

The problem for Mr. Biden is no other debate is scheduled until Sept. 10, meaning he has no obvious opportunity to recover for months. And as the veteran Democratic strategist said, this was not like Mr. Obama losing to Mitt Romney in 2012, which was a tactical setback. “This is existential,” the strategist said.

So, rather than resetting the campaign in Mr. Biden’s favor, as it had anticipated, the president’s team ended the evening knowing that the task of the next few days if not weeks would be to stem the damage and rally the party behind their beleaguered leader.