Dozens of congressional Republicans disowned their party’s presidential nominee over the weekend, saying a decade-old tape of lewd talk about women has cost him the ability to lead and are trying to mount a write-in campaign for Indiana Gov. Mike Pence instead.

The mass desertion comes at a critical time for Mr. Trump and Republican lawmakers, who are facing an ever-tougher map as they strive to keep control of Congress.

Vulnerable Republicans raced for the exits Saturday and Sunday, saying they had overlooked so much for the sake of party unity but that Mr. Trump’s comments were too much for them to stomach.

“While I continue to respect those who still support Donald Trump, I can no longer support him,” said Sen. Rob Portman, an Ohio Republican who is up for re-election. “I continue to believe our country cannot afford a Hillary Clinton presidency. I will be voting for Mike Pence for president.”

Rank-and-file voters, though, were less eager to toss Mr. Trump overboard. A snap poll taken the day after the comments surfaced Friday showed that just 12 percent of Republican voters want Mr. Trump to quit, while 74 percent of Republicans want their party to stick by the billionaire businessman.

Former New York Mayor Rudolph W. Giuliani came to Mr. Trump’s defense, saying the New York businessman’s remarks were lewd but not outside of the scope of male banter behind closed doors.

“The fact is that men at times talk like that,” Mr. Giuliani said on CNN’s “State of the Union.” “Not all men, but men do. He was wrong for doing it. I am not justifying it. I believe it’s wrong. I know he believes it’s wrong. I believe that this is not the man that we’re talking about today.

“Gosh almighty, he who is without sin here throw the first stone,” he added.

Republican women began to circle the wagons after watching big names in the party make their break.

“Those that are jumping ship are establishment GOP that never supported him in the first place,” said Minnesota state Sen. Carrie L. Rudd.

The comments that have sparked the Trump rethink came on a tape released Friday by The Washington Post, with Mr. Trump bragging over a live microphone to television show host Billy Bush about attempting to seduce women, including one unidentified married celebrity.

“And when you’re a star, they let you do it,” Mr. Trump says in the tape, using vulgarisms to refer to sex and women’s genitalia. “You can do anything.”

Mr. Trump released an apology late Friday saying it was “locker room” talk. He said former President Bill Clinton, husband of Democratic nominee Hillary Clinton, said even worse things.

By the weekend, though, Mr. Trump struck a more defiant note, blasting the press and Republican leaders who were abandoning him.

“So many self-righteous hypocrites. Watch their poll numbers — and elections — go down,” he said in a Twitter post Sunday morning.

Mr. Trump also took solace in a Los Angeles Times/University of Southern California poll that showed him still in the lead in a matchup with Mrs. Clinton. That poll has been consistently favorable to Mr. Trump, while most other surveys show him trailing the Democratic nominee.

The danger for Republicans is that a Trump collapse could threaten their hopes of keeping control of Congress.

Democrats are eager to capitalize, calling the leaked tape a defining moment for the Republican Party and insisting House, Senate and gubernatorial candidates pick sides between Mr. Trump and his Republican detractors.

The Democratic National Committee said Republican lawmakers stood by Mr. Trump when he mocked a disabled reporter, accused a federal judge of bias because of his Mexican heritage and verbally attacked a Muslim family whose son, an Army captain, died while serving in Iraq.

“Republicans: You own this,” the DNC said in a public memo Sunday.

As of Sunday, 16 Republican senators had called for Mr. Trump to drop out of the race, as had some two dozen members of the House.

House Speaker Paul D. Ryan, Wisconsin Republican, disinvited Mr. Trump from a planned joint appearance Saturday, but he has not joined the calls for the nominee’s ouster. Neither has Senate Majority Leader Mitch McConnell, Kentucky Republican, though he called Mr. Trump’s words “repugnant.”

Mr. Pence has emerged as a focal point for the dump-Trump crowd. Mr. Portman, Sen. Kelly Ayotte — an embattled Republican from New Hampshire — and Sen. Cory Gardner, a Colorado Republican whose next election is 2020.

“I will not vote for Donald Trump,” said Mr. Gardner, who in August announced that he would vote for the “entire Republican ticket in November” without naming the real estate mogul. “If Donald Trump wishes to defeat Hillary Clinton, he should do the only thing that will allow us to — step aside, and allow Mike Pence to be the Republican Party’s nominee.”

Mr. Pence is fresh off a well-received performance last week in the vice presidential debate, having survived a barrage of attacks from Sen. Tim Kaine, Mrs. Clinton’s running mate, as well as intense prodding by the moderator, CBS News reporter Elaine Quijano.

Over the weekend, Mr. Pence gave no signal that he was thinking of dropping off the ticket, nor did he encourage the pleas from his former colleagues in Congress to cast ballots for him as a write-in candidate.

Party tacticians said with early voting beginning, it’s logistically futile to think that Mr. Trump could be replaced, even by Mr. Pence.

• Ralph Z. Hallow contributed to this report.

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