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Mr. Pence said in a statement that he was offended by the words and actions described by Donald Trump in the video, and cast Mr. Trumps second debate with Hillary Clinton, on Sunday, as an urgent moment to turn around the campaign.

In a 2005 recording obtained by The Washington Post, Donald J. Trump, the Republican presidential nominee, talks about women in vulgar terms to Billy Bush, then the host of Access Hollywood.

I do not condone his remarks and cannot defend them, Mr. Pence said, adding, We pray for his family and look forward to the opportunity he has to show what is in his heart when he goes before the nation tomorrow night.

By Saturday evening, no fewer than 36 Republican members of Congress and governors who had not previously ruled out supporting Mr. Trump disavowed his candidacy, an unprecedented desertion by the institutional Republican Party of its own standard-bearer just a month before Election Day.

The growing wall of opposition recalled the determination of the party establishment this year to deny Mr. Trump the nomination in the first place. He easily swatted away that effort, but Mr. Trump now finds himself in a far more precarious state. Facing a more vast and diverse electorate, his lightly organized campaign was already listing before the videotape was released.

Aides described Mr. Trump as shaken, watching news coverage of the video with a mix of disbelief and horror. Shortly after midnight, he had released a videotaped statement, saying, Ive said and done things I regret, and the words released today on this more than a decade-old video are one of them.

In a brief telephone interview on Saturday, he shrugged off the calls to leave, saying he would never drop out of this race in a million years.

I havent heard from anyone saying I should drop out, and that would never happen, never happen, Mr. Trump said. Thats not the kind of person I am. I am in this until the end.

Far from sounding rattled, Mr. Trump insisted that he could still prevail in November.

Oh, yeah, we can win we will win, he said. We have tremendous support. I think a lot of people underestimate how loyal my supporters are.

A couple of hours later, the campaign released a statement from his wife, Melania. The words my husband used are unacceptable and offensive to me, she said. This does not represent the man that I know.

I hope people will accept his apology, as I have, and focus on the important issues facing our nation and the world, she said.

But the situation had grown so dire that many in the party were all but pleading with him to withdraw and let Mr. Pence serve as the presidential nominee. On Saturday afternoon, Senator John Thune of South Dakota, the chairman of the Republican Conference, became the most senior Republican to call on Mr. Trump to make way for Mr. Pence.

The exodus began late Friday night when a handful of Utah Republicans who said they would support Mr. Trump indicated that they could no longer tolerate their nominee.

But it was not until a pair of conservative women, Representatives Barbara Comstock of Virginia and Martha Roby of Alabama, implored Mr. Trump to withdraw that previously hesitant Republicans stepped forward to reject Mr. Trumps candidacy.

Kelly Ayotte of New Hampshire was the first Republican senator facing a competitive re-election to say she would no longer back Mr. Trump, announcing in a statement that she would write in Mr. Pence for president instead.

Im a mom and an American first, and I cannot and will not support a candidate for president who brags about degrading and assaulting women, she wrote on Twitter.

Ms. Ayotte was joined just hours later by Mr. McCain, who is also running for re-election, and Representative Joe Heck of Nevada, who is locked in a close race for the Senate seat now held by Harry Reid, the Democratic minority leader, who is retiring.

It was an admission that Mr. Trump now posed an immediate threat to their own candidacies and that, to have any chance to survive, they had to risk angering his ardent supporters. At a party gathering on Saturday in Wisconsin, Speaker Paul D. Ryan, who had disinvited Mr. Trump and said he was sickened by the video, was greeted with a few boos, and Mr. Heck was both jeered and applauded when he announced to a crowd in Nevada that he was not backing the presidential nominee.

Mr. Ryan told his crowd he would not be discussing the elephant in the room, the 2005 video showing a bus that had Mr. Trump aboard, and included an audio recording of him privately bantering with other men.

Mr. Trump, then newly married to Ms. Trump, crassly boasted about groping womens genitals, vulgarly commented on their bodies and generally described women as sex objects who could not resist his advances.

In his video statement released early Saturday, Mr. Trump said: Anyone who knows me knows these words dont reflect who I am. I said it, I was wrong, and I apologize.

I pledge to be a better man tomorrow and will never, ever let you down, he added, before ending the message with a promise to bring up the sex scandals of Bill Clintons presidency and Hillary Clintons response to them.

Inside Trump Tower, though, Mr. Trumps defiant public responses belied the reality of a 24-hour period in which he was alternately angry and distressed, according to two people with direct knowledge of his behavior who were granted anonymity to discuss private conversations.

Hillary Clinton has an 83% chance of winning the presidency.

Mr. Trump and his son-in-law, Jared Kushner, initially expressed skepticism upon hearing that such a recording existed, saying those comments did not sound like him. When Mr. Trump heard the tape played, he acknowledged it was him, but he believed the fallout would not be dramatic.

Mr. Pence, however, was dismayed, and called into Trump headquarters on Friday night to urge Mr. Trump to apologize.

On Saturday morning, Mr. Pence called Mr. Trump and told him he had to handle the next 48 hours alone because he did not think he would be an effective surrogate.

Mr. Trump, after monitoring television coverage, realized he was becoming isolated by his party.

Mr. Trumps aides did not explicitly ask top advisers and allies to do their usual defense of Mr. Trumps comments, according to one person briefed on the discussions, but they did ask people to stand by his side. A few supporters did, including Ben Carson; the conservative radio host Laura Ingraham; and Robert and Rebekah Mercer, the wealthy father and daughter who are perhaps Mr. Trumps most important backers, and who said in a statement that they considered the video locker room braggadocio.

America is finally fed up and disgusted with its political elite, they said. Trump is channeling this disgust, and those among the political elite who quake before the boombox of media blather do not appreciate the apocalyptic choice that America faces on Nov. 8.

Two of Mr. Trumps most prominent supporters Gov. Chris Christie of New Jersey and former Mayor Rudolph W. Giuliani of New York went to Trump Tower around noon to huddle with Mr. Trump and try to get in some debate preparation.

In the afternoon, more damaging news hit the web and cable television, with a CNN report on the numerous lewd and tasteless comments he had made over the years on The Howard Stern Show.

CreditStephen Crowley/The New York Times

Just before 5 p.m., Mr. Trump emerged for about five minutes, briskly striding through his gilded lobby to a waiting crowd of supporters on the sidewalk. He pumped his right fist in the air as fans surrounded him.

Hundred percent, Mr. Trump told reporters who yelled questions about whether he would stay in the race.

Mr. Pence flew from Indianapolis to a fund-raising event in Rhode Island, where he told supporters that the election was about more than one man, said Joseph A. Trillo, the chairman of Mr. Trumps campaign in the state. Over hors doeuvres in a Newport mansion, Mr. Pence offered a pep talk without making direct mention of the days dire events.

He used the terminology that its a movement and its bigger than Donald Trump, Mr. Trillo said.

At the same time, leading Republicans were demanding that the Republican National Committee, which has been helping the Trump campaign financially and organizationally, abandon Mr. Trump and turn its attention to salvaging other candidates down the ballot.

Representative Charlie Dent, Republican of Pennsylvania, said the committee should no longer defend the indefensible.

He called on Reince Priebus, the party chairman, to force Mr. Trump off the ticket or face the consequences.

The chairman of the R.N.C. must look out for the good of the party as a whole, so he should be working to get him to step down, Mr. Dent said. If he cant, then he should step down.

The committee remained silent on Saturday as members of Congress began fleeing from Mr. Trump, not responding to news media inquiries and, senior Republican officials said, not coordinating with other campaign organizations. However, one senior Republican official said Mr. Priebus was deeply distressed. He went to Trump Tower in the afternoon to talk to Mr. Trump.

Powerful donors and business interests signaled that they would redirect their attention to down-ballot candidates. Republican power brokers had hoped until recently that Mr. Trump might make a credible showing in the presidential election, aiding the party in its other crucial races.

But Republicans now say that their worst fears have come to pass, as Mr. Trump has unraveled in a series of missteps after his first debate with Mrs. Clinton.

Even before Mr. Trumps 2005 comments came to light, internal Republican polling showed him losing ground among three groups that had long been wary of his candidacy: independents, women, and voters with college degrees.

That slide is likely to accelerate now, Republicans said, potentially sending voters fleeing toward Democrats or convincing them that they should stay home on Nov. 8. Either outcome would be ruinous for Republican candidates beyond the presidential race.

It will be difficult in the extreme for him to recover from this, but the biggest impact is likely to be its effect on all the down-ballot races, said Fred Malek, the finance chairman of the Republican Governors Association, who called Mr. Trumps comments beyond disgusting.

If they pull the plug on support for Trump, he said, the vast majority of voters will certainly understand that and most will respect it.

Jonathan Martin reported from Washington, and Alexander Burns and Maggie Haberman from New York. Ashley Parker and Patrick Healy contributed reporting from New York, and Thomas Kaplan from Indianapolis.

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A version of this article appears in print on October 9, 2016, on page A1 of the New York edition with the headline: Lewd Trump Tape Is Breaking Point for Many in G.O.P.

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