Burton claimed Affleck, 45, groped her when she was a host of MTV's TRL in 2003.
The incident resurfaced after Affleck condemned the sexual harassment allegations against Hollywood mogul Harvey Weinstein saying he was "saddened and angered" by claims against him.
"What I was accused of by a woman was touching her breast while I gave her a hug. I don't remember it, but I absolutely apologise for it. I certainly don't think she's lying or making it up," Affleck told Stephen Colbert on his "Late Show".
The "Justice League" actor also said it was "awful" to find out about the extent of the "terrible crimes" committed by Weinstein.
I thought I understood the extent of the problem, and I thought I had a handle on it," Affleck said.
"The truth is, I really didn't. I didn't understand what it was to be groped, to be harassed, to be interrupted, talked over, paid less, pushed around, belittled, all of the things that women deal with that for me as a man I have the privilege of not having to deal with," he added.
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