The actor was accused of "grabbing" Burton's breasts when they appeared together on American TV show TRL in 2003.
Batman star Affleck faced a barrage of criticism on social media after he commented on the sexual allegations made against producer Harvey Weinstein from a number of women in the film industry.
The groping claim involving Burton resurfaced when a Twitter user wrote, in response to a tweet about Affleck's statement: "He also grabbed Hilarie Burton's breasts on TRL once. Everyone forgot though."
The Twitter user said they were "so sorry that happened" to Burton, to which the actress replied: "Seriously, thank you for that. I was a kid."
Burton also shared a video snippet that appeared to show the incident and wrote: "Girls. I'm so impressed with you brave ones.
Affleck tweeted, hours after her online posts: "I acted inappropriately toward Ms. Burton and I sincerely apologize."
He wrote: "I am saddened and angry that a man who I worked with used his position of power to intimidate, sexually harass and manipulate many women over decades.
"The additional allegations of assault that I read this morning made me sick. This is completely unacceptable, and I find myself asking what I can do to make sure this doesn't happen to others. We need to do better at protecting our sisters, friends, co-workers and daughter.
Weinstein has denied many of the allegations and issued a vehement denial over three allegations of rape which emerged on Tuesday.