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.
The latest to withdraw their support are former Republican presidential candidate John McCain and former Secretary of State Condoleezza Rice.
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.
Mr Trump himself stressed that there was "zero chance I'll quit", adding that he was getting "unbelievable" support.
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House Speaker Paul Ryan had originally invited Mr Trump to attend a campaign event in Wisconsin this weekend but rescinded his invitation, saying he was "sickened" by what he had heard.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".