By Jonathan Martin & Alexander Burns, New York Times–

WASHINGTON — Donald J. Trump’s campaign was teetering on Saturday after the release of a video in which he speaks of women in vulgar sexual terms, with more Republican leaders calling for him to leave the ticket and demanding that the party shift focus to down-ballot races.

Representative Charles Dent, Republican of Pennsylvania, said on Saturday that the party, which has been helping the Trump campaign financially and organizationally, should no longer “defend the indefensible.” He urged the party to abandon Mr. Trump if he refuses to withdraw and focus solely on electing the rest of the ticket.

“The priority for the Republican Party has to be protecting our congressional majorities,” he said.

In an extraordinary statement from a running mate, Gov. Mike Pence of Indiana indicated that he still supported Mr. Trump, but that he was highly displeased with what he heard on the video.

“As a husband and father, I was offended by the words and actions described by Donald Trump in the 11-year-old video released yesterday,” Mr. Pence said. “I do not condone his remarks and cannot defend them.”

Referring to Mr. Trump’s videotaped apology early Saturday morning, and the critical debate coming Sunday night, Mr. Pence continued: “I am grateful that he has expressed remorse and apologized to the American people. 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.”

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

“I haven’t heard from anyone saying I should drop out, and that would never happen, never happen,” Mr. Trump said. “That’s not the kind of person I am. I am in this until the end.”

Far from sounding shaken, Mr. Trump insisted that he believed he could still win the presidency 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.”

Mr. Trump said he was going ahead with preparations for Sunday night’s debate and that he had not decided whether to make a public appearance later Saturday or on Sunday morning to address his remarks about women. He added that he did not feel rattled by the fierce criticism over his comments.

“I’m doing fine — focusing on the debate, getting ready, focusing on talking to voters,” Mr. Trump said. “I’ll be fine.”

The video released on Friday and recorded in 2005 showed 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 his third wife, Melania, crassly boasted about groping women’s genitalia, vulgarly commented on their bodies and generally described women as sex objects who could not resist his advances.

In a response released on video just after midnight Saturday, Mr. Trump said: “Anyone who knows me knows these words don’t 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 on a defiant note, promising to bring up the sex scandals of Bill Clinton’s presidency and Hillary Clinton’s response to them.

In his interview on Saturday, Mr. Trump said that no Republican leaders or officials had suggested to him that he rethink his candidacy. And people close to the Trump campaign and the Republican National Committee roundly denied that there was any effort afoot to ease him from the race.

But for many in the party, it amounted to a breaking point, and a number of its elected officials publicly called on Mr. Trump on Saturday to step aside. The comments marked the last humiliation they were willing to suffer after months of being made to disavow or play down remarks about race and gender from their nominee that were as inflammatory as they were routine.

“This can’t continue,” said Senator Mike Lee of Utah, appearing anguished in a video he posted on Facebook. “Mr. Trump, I respectfully ask you with all due respect to step aside. Step down.”

In Alabama, a deeply conservative state, some elected Republicans said they could no longer support Mr. Trump and implored him to leave the race.

“Donald Trump’s behavior makes him unacceptable as a candidate for president, and I won’t vote for him,” said Representative Martha Roby of Alabama, adding, “Hillary Clinton must not be president, but with Trump leading the ticket, she will be.”

And even Senator Shelley Moore Capito of West Virginia, one of Mr. Trump’s strongest states, said she was “taken aback” by the nominee’s behavior, adding that he should consider what is best for the party. “The next step may be for him to re-examine his candidacy,” Ms. Capito said in an interview on Saturday.

Jim Hobart, a Republican pollster, said the disclosure of Mr. Trump’s 2005 comments would hit him especially hard with a critical group that already largely opposes him: suburban women. He and other Republicans said the new uproar placed immense pressure on Mr. Trump to demonstrate on Sunday, in his second debate with Mrs. Clinton, that he is capable of true contrition and the sort of presidential temperament that most Americans believe he lacks.

“Tomorrow’s debate performance is everything,” Matt Borges, the chairman of the Ohio Republican Party, said Saturday. “If there’s a chance to recover, that’s it.”

But many in the party have concluded that Mr. Trump is a surefire loser and now only hope that he will not further imperil Republican prospects when he meets Mrs. Clinton on Sunday night in St. Louis.

“It’s important not just for his sake to have a better debate in order to stabilize his campaign, but also to mitigate any damage down-ballot,” Mr. Dent said.

Mr. Dent, who spent Friday with former President George W. Bush at a fund-raiser for the endangered Pennsylvania Senator Patrick J. Toomey, said he hoped Mr. Trump would not inflame his own controversy by using the debate to invoke Mr. Clinton’s infidelity. “In my view, that’s not a wise strategy,” he said.

Yet holding up Mr. Clinton’s sexual affairs is precisely what Mr. Trump did at the end of the apology video he was urged to record by panicked aides and party officials. And he vowed to keep it up.

“Bill Clinton has actually abused women and Hillary has bullied, attacked, shamed and intimidated his victims,” Mr. Trump said. “We will discuss this more in the coming days.”

Others in the party, though, expressed hope late Friday and Saturday that they would no longer have to discuss Mr. Trump much in the coming days. Representative Barbara Comstock of Virginia, facing a competitive re-election in an affluent suburban Washington-area district, excoriated her party’s standard-bearer and urged him to withdraw his candidacy.

“This is disgusting, vile and disqualifying,” Ms. Comstock said in a statement, adding that Mr. Trump “should step aside and allow our party to replace him with Mike Pence or another appropriate nominee from the Republican Party.”

Another House Republican facing a difficult re-election, Representative Mike Coffmann of Colorado, also demanded that Mr. Trump bring an end to his campaign, calling the nominee’s defeat “almost certain.” And an array of Utah Republicans, including Gov. Gary Herbert and Representative Jason Chaffetz, withdrew their support, raising the once-unthinkable prospect that a Republican nominee for president might have trouble carrying the heavily conservative and Mormon state.

It is unclear that even those stern rebukes of Mr. Trump will be enough to spare other Republicans political backlash. Democratic candidates and lawmakers have insisted that such a late rejection of Mr. Trump amounts to a hollow gesture, given the mountain of crude and offensive pronouncements Mr. Trump has offered since announcing for president in July 2015.

“When will elected Republicans, especially Republican women, quit apologizing for this miserable excuse for a leader?” said Senator Claire McCaskill, Democrat of Missouri. “I’m confident they also find this disgusting.”

Many Republicans were closely watching the actions of two elected party members who face competitive races this year and hold influential platforms in the party. Senators Kelly Ayotte of New Hampshire, one of the most prominent women in the party, and John McCain of Arizona, the former P.O.W. and presidential nominee, both sharply criticized Mr. Trump for his language.

On Saturday morning, Ms. Ayotte went further.

“I’m 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. She said she would write in Mr. Trump’s running mate, Gov. Mike Pence of Indiana, for president on her ballot.

Mr. Trump was already plummeting in the polls last week, unnerving Republicans who were watching independent voters flee from his candidacy after his disastrous first debate was worsened by a prolonged diatribe about the weight of a beauty queen.

He has prepared more rigorously for the second debate, allowing Gov. Chris Christie of New Jersey to pepper him with questions designed to unnerve him — but nothing as jarring as the explosive disclosure of the graphic audiotape.

Betty D. Montgomery, a former Republican attorney general in Ohio, predicted Mr. Trump’s comments would make “a huge difference” in dissuading her state’s remaining undecided female voters from supporting his candidacy.

“It just makes me so angry and more certain of where I am,” said Ms. Montgomery, who previously said she would not support Mr. Trump.

Jonathan Martin reported from Washington, and Alexander Burns from New York. Maggie Haberman and Patrick Healy contributed reporting from New York, and Tom 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 A Breaking Point For Many in G.O.P. Order Reprints| Today’s Paper|Subscribe