Mr Alexander "unreservedly" apologised for "crude and inappropriate" jokes about rape made in a video posted to YouTube in 2011.
Video footage shows the former tennis great turned MP telling jokes at a Brisbane pub in the 1990s.
Mr Alexander opens one of the jokes with: "A black guy in Chicago is a witness to a rape..."
In a statement, Mr Alexander said there was "no place" for the jokes.
“More than twenty years ago I told crude and inappropriate jokes, which were completely unacceptable and I apologise unreservedly. There is no place for jokes about violence against women. Again, I apologise unreservedly," he said.
He doubled down on his apology, telling reporters in Sydney on Thursday he was "horrified" after seeing the footage.
"When I saw them this morning, I was horrified at my language and I apologise," he said.
Mr Alexander is up against former NSW premier Kristina Keneally in the December 16 ballot, but he admits the negative publicity will damage his campaign.
"I think that there is no doubt that it is damaging. I have worked very, very hard, and those people who have gotten to know me over the last seven years and who trust me, and believe that I have worked in their best interest, has been a loyalty hard won," he said.
"I hope that they will find it in their hearts to forgive me and accept my apology."
When asked on his thoughts regarding the timing of the video's surfacing - weeks before the ballot - Mr Alexander said it was "up for others to speculate on".
"I can only apologise for my behaviour, for my bad judgement. And for the inappropriateness of those jokes," he said.
"There will be some people who will find it difficult to accept my apologies and forgive, and I have to accept that.
"There will be others who will accept the apology and allow me to get on with the work that I've been doing here in this electorate for the last seven years."
Malcolm Turnbull said it is a measure of John Alexander's dignity that the Bennelong by-election candidate has apologised for making racially insensitive and rape jokes 20 years ago.
Asked about the video, the prime minister welcomed Mr Alexander's apology and said such ill-judged statements, whether intended as jokes or not, are completely unacceptable.
"There is no place for joking about violence against women," Mr Turnbull told reporters in Canberra.
"It is a measure of the man and of the dignity of the man that he has acknowledged that those remarks were unacceptable and he's unreservedly apologised for them."
