His comments, captured in a video, concern the rape and murder of an Australian missionary in 1989 in Davao.
The footage shows Rodrigo Duterte make fun of the assault by prison inmates on the missionary when he was town mayor.
His joke appeared to make fun of her death, saying she was "so beautiful" that "the mayor should have been first".
Current President Benigno Aquino's spokesperson, Herminio Coloma, said the comments reflected Mr Duterte's "lack of fitness for the presidency" and his "utter lack of respect for women".
Disgusting endorsement of sexual violence by #Philippines pres contender Duterte should be utterly condemned @hrw https://t.co/8zwoGZAADl
— Phil Robertson (@Reaproy) April 17, 2016
Rival presidential candidate, Vice-President Jejomar Binay, added: "You are a crazy maniac who doesn't respect women and doesn't deserve to be president."
He says that he will not apologise for his comment, saying: "I will not. I am even willing to lose the presidency."
Speaking to the BBC, Human Rights Watch Deputy Asia Director Phil Robertson said the comments "were a disgusting endorsement of sexual violence."
Last year Human Rights Watch called Rodrigo Duterte the death squad mayor for the way he dealt with violent crime, when more than 1,000 people were killed.
Reports from the time name the Australian missionary as 36-year-old Jacqueline Hamill.
Four other hostages were killed in the incident.
Filipinos go to the polls in May.
