The tape, released during the 2016 presidential campaign, revealed the now-infamous "grab them by the pussy" comments made by Mr Trump that led several major Republican figures to withdraw their support for the candidate.
In an op-ed published in The New York Times, Bush said: "Of course he said it."
Mr Trump was sitting with Bush, who at the time was an Access Hollywood host, when he bragged in detail about trying to pursue an unnamed married woman, then turned his attention to an actress on the set.
"You know I'm automatically attracted to beautiful — I just start kissing them," Mr Trump said.
"It's like a magnet. Just kiss. I don't even wait.
"And when you're a star they let you do it. You can do anything.
"Grab them by the pussy. You can do anything."
In Bush's op-ed, he said everyone nearby at the time thought they were listening to "a crass stand-up
Bush said he felt the need to write the piece following reports Mr Trump had recently suggested in private the tape was not authentic.
"President Trump is currently indulging in some revisionist history," Bush wrote.
Bush, who was an NBC television host until he was fired due to his enabling role in the Access Hollywood video, said the story "has hit a raw nerve."
"I can only imagine how it has reopened the wounds of the women who came forward with their stories about him, and did not receive enough attention."
The tape's release prompted more than a dozen women to come forward to say Mr Trump had sexually assaulted or harassed them.
Mr Trump and his administration deny the allegations.
In the wake of recent sexual assault allegations against prominent news media and entertainment figures, such Harvey Weinstein, Charlie Rose, Kevin Spacey, and Louis CK, Bush said he hopes the hard work of exposing such injustices is over.
Bush also detailed the allegations made by two women, Jill Harth and Kristin Anderson, who accused the US President of sexual assault.
Bush is the first cousin of former US President George W Bush, and said he believes Ms Harth and Ms Anderson are telling the truth.
At the time of the tape's release, Mr Trump's wife Melania said he had been, "egged on" by Bush to, "say dirty and bad stuff".
Media player: "Space" to play, "M" to mute, "left" and "right" to seek.
VIDEO: Movie icon George Clooney links sexual assault in Hollywood to the behaviour of Trump and Fox News personalities.(ABC News)