“There is nothing that will cause his dropping out,” Mr. Giuliani said. “That is wishful thinking of the Clinton campaign and those who have opposed him for a long time.”

Though two of his biggest financial backers are sticking with him ...

Robert and Rebekah Mercer, the Republican megadonors who previously supported Senator Ted Cruz, suggested Mr. Trump’s remarks were irrelevant to the race.

“If Mr. Trump had told Billy Bush, whoever that is, earlier this year that he was for open borders, open trade and executive actions in pursuit of gun control, we would certainly be rethinking our support for him,” they said in a statement.

The Mercers also compared Mr. Trump’s actions to Bill Clinton’s treatment of women. The tactic is a favorite of Mr. Trump’s supporters, including Sean Hannity, the Fox News host, who devoted most of his show on Friday evening to allegations of abuse by Mr. Clinton.

... many prominent Republicans have rescinded endorsements and distanced themselves from him.

• Condoleezza Rice, the secretary of state under President George W. Bush, who has generally has not discussed politics since becoming a private citizen, called for Mr. Trump to leave the race. She said she hoped to support someone who has the dignity and stature to be president.

“Enough!” Ms. Rice said in a Facebook post. “Donald Trump should not be President. He should withdraw.”

• Mr. McCain, who has criticized Mr. Trump repeatedly, officially withdrew his backing. “I have wanted to support the candidate our party nominated,” he said in a statement. He added: “But Donald Trump’s behavior this week, concluding with the disclosure of his demeaning comments about women and his boasts about sexual assaults, make it impossible to continue to offer even conditional support for his candidacy.”