At least a dozen Republicans have said they will not be voting for the New York billionaire, since the comments emerged on October 7.
Donald 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.
John McCain said Donald Trump’s comments “make it impossible to continue to offer even conditional support for his candidacy”, while Condoleeza Rice said: “Enough! Donald Trump should not be President. He should withdraw.”
Another senior Republican, 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.”
Kelly Ayotte – who faces a competitive race for re-election – said she would not vote for Hillary Clinton but instead would “write in” Mike Pence, Donald Trump’s vice-presidential running mate, on her ballot paper.
Several other Republicans also said they would vote for Mike Pence.
Donald Trump himself stressed that there was “zero chance I’ll quit”, adding that he was getting “unbelievable” support.
In a tweet, the Republican candidate said: “The media and establishment want me out of the race so badly.”
Donald Trump’s wife, Melania, issued a statement on October 8 saying: “The words my husband used are unacceptable and offensive to me.”
Melania Trump said her husband had “the heart and mind of a leader”.
Mike Pence said he was “offended” by Donald Trump’s video, but grateful he had expressed remorse and apologized to the American people.
“We pray for his family,” he said in a statement.
House Speaker Paul Ryan had originally invited Donald Trump to attend a campaign event in Wisconsin this weekend but rescinded his invitation, saying he was “sickened” by what he had heard. Mike Pence was due to go in his running mate’s place, but declined to attend.
Meanwhile, Hillary Clinton called Donald Trump’s comments in the tape “horrific”.
In the recorded comments, which date back to 2005 when Donald Trump was appearing as a guest on a soap, he says “you can do anything” to women “when you’re a star”.
Donald Trump released a video statement apologizing for the comments.
His 2005 comments, posted by the Washington Post, overshadowed the release of transcripts of Hillary Clinton’s speeches to private events, by the WikiLeaks.
Donald Trump had married his third wife Melania a few months before the recording. She said on October 8: “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 Donald Trump and Hillary Clinton will take place on Sunday evening, October 9, in St Louis.
Donald Trump recently said he would not bring up stories about Bill Clinton’s infidelities in the debate, after previously threatening to do so.
However, 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.”
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