At least a dozen Republicans have said they will not be voting for him, since the comments emerged on Friday.
Mr Trump says he will never drop out of the race to be president and will never let his supporters down.
He has been under pressure after a tape from 2005 of him bragging about groping and kissing women was broadcast.
Condoleezza Rice: "Donald Trump should not be president."
Mr McCain said Mr Trump's comments "make it impossible to continue to offer even conditional support for his candidacy", while Ms Rice said: "Enough! Donald Trump should not be President. He should withdraw."
New Hampshire Senator Kelly Ayotte said in a statement: "I cannot and will not support a candidate for president who brags about degrading and assaulting women," she said
Ms Ayotte - who faces a competitive race for re-election - said she would not vote for Mrs Clinton but instead would "write in" Mike Pence, Mr Trump's vice-presidential running mate, on her ballot paper.
Several other Republicans also said they would vote for Mr Pence.
And in a tweet, the Republican candidate said "the media and establishment want me out of the race so badly".
Mr Trump's wife Melania issued a statement on Saturday saying: "The words my husband used are unacceptable and offensive to me."
She said her husband had "the heart and mind of a leader".
Mr Pence said he was "offended" by Mr Trump's video, but grateful he had expressed remorse and apologised to the American people.
"We pray for his family," he said in a statement.
House Speaker Paul Ryan had originally invited Mr Trump to attend a campaign event in Wisconsin this weekend but rescinded his invitation, saying he was "sickened" by what he had heard. Mr Pence was due to go in his running mate's place, but declined to attend.
Meanwhile, Hillary Clinton, Mr Trump's Democratic election rival, called his comments in the tape "horrific".
Warning: The rest of this article contains some graphic language
In the recorded comments, which date back to 2005 when Mr Trump was appearing as a guest on a soap, he says "you can do anything" to women "when you're a star" and is heard saying "grab them by the pussy".
Mr Trump's 2005 comments, posted by the Washington Post, overshadowed the release of transcripts of Mrs Clinton's speeches to private events, by the whistle-blowing site Wikileaks.
The candidate had married his third wife Melania a few months before the recording. She said on Saturday: "I hope people will accept his apology, as I have, and focus on the important issues facing our nation and the world."
The second TV debate between Mr Trump and Mrs Clinton will take place on Sunday evening in St Louis.
Mr Trump recently said he would not bring up stories about Bill Clinton's infidelities in the debate, after previously threatening to do so.
But in his video apology, he attacked the former president directly:
"Bill Clinton has actually abused women, and Hillary has bullied, attacked and shamed his victims.
"We'll discuss this in the coming days," he said. "See you at the debate on Sunday."
Is this Trump row different? - Katty Kay, Presenter, BBC World News
Former Miss Universe Alicia Machado has spoken out against Mr Trump
Enough, quite possibly, to sink any chance he had of winning the White House.
There is a violence in the phrases "grab 'em by the pussy" and "you can do anything" that any victim of abuse would recognise and that most women would find sickening.
But this tape doesn't just offend women, judging from the reaction in the Republican party. It has offended a lot of men too. Whether those men will now withdraw their endorsements of him is yet to be seen.