The man was bleeding from the nose and lip as he was dragged down the aisle by personnel the airline identified as "law enforcement."
United said in a statement after the episode that it needed to bump passengers because the flight was overbooked.
According to one account, the airline offered $400 for volunteers; having gotten no takers, it increased the offer to $800. When no one gave up their seat, United chose four passengers at random to get off the flight.
Security was called in, the airline said, when one passenger refused to leave his seat.
Videos posted to social media show passengers protesting as the man is wrestled from his seat and dragged down the aisle.
One passenger asks whether the airline can rent a car so its employees could get to Louisville.
"Good going, guys," one passenger says to the officers. "You busted his lip," another man says. "Oh, my God! Look at what you did to him!" a third passenger cries.
Audra Bridges, one of those on the plane who shared a video online, urged people to share it and wrote, "We are all so disgusted."
United's statement did say it was sorry -- but not for its treatment of the passenger in the videos.
"We apologize for the overbook situation," the statement said.