Ford said on his weekly radio show on Newstalk 1010, however, he will not abdicate his office, the Canadian Broadcasting Corp. reported.
The mayor called on police Chief Bill Blair to release the video "for every single person in this city to see. "
"That is the right thing to do and chief, I am asking you to release this video now," Ford said.
"Whatever this video shows ... people need to judge for themselves what they see on this video."
Last week, Blair said footage on the video investigators retrieved from a seized hard drive was "consistent" with reports that it shows the mayor smoking from a crack pipe.
Ford, who has not been charged with any crime, denied using crack cocaine and the video's existence when the reports first emerged in May. On Sunday, he said he is "the first one to admit, I am not perfect."
"I have made mistakes ... and all I can do right now is apologize for the mistakes," he said, adding later in the program, "I've made mistakes, like, where do I begin? For example, the Danforth, that was pure stupidity. I shouldn't have got hammered down at the Danforth. If you're going to have a couple of drinks, you stay at home and that's it, you don't make a public spectacle of yourself."
He also acknowledged having gotten "a little out of control" on St. Patrick's Day.
"I'm going to do my very best to make sure these mistakes don't happen again. And I don't know what else to say," he said.
"I am going to ride the storm out and just keep doing what I was elected to do."
Ford's approval rating rose despite the crack cocaine allegations, a poll found.
A Forum Research survey released at the end of the week showed his approval rating jumped 5 percent -- from 39 percent to 44 percent -- even though 98 percent of the respondents were aware of the allegations.
"Remember we are talking about approval rating, and that's a question pertaining to his job as mayor," Forum Research President Lorne Bozinoff told CBC News. "As long as these allegations are not impinging on his ability to do his job, people are satisfied."
Bozinoff said it was possible some of the added support came from voters who were sympathetic to Ford's belligerent response to media crews who descended on his home after police released the video -- which Ford had consistently maintained did not exist.
At the same time, however, a majority of respondents believed Ford would have to resign.
"It's not like he's got a free pass on this," Bozinoff said.
The Forum Research was conducted late Thursday after a police news conference on the alleged incident. A total of 1,032 Toronto residents were surveyed with a margin of error of 3 percentage points.