Oscar Munoz said he felt "shame and embarrassment" and vowed it would never happen again to a seated passenger on one of United's aircraft.
The embattled aviation boss said the passenger in question, David Dao, deserved "certainly an apology".
Mr Munoz initially described Mr Dao as "disruptive and belligerent".
United Airline's public relations disaster
Why do airlines overbook?
"That shame and embarrassment was pretty palpable for me and for a lot of our family," the contrite chief executive told ABC's Good Morning America programme.
An online petition calling for Mr Munoz to resign had gained more than 60,000 signatures by Wednesday afternoon.
But when asked if he would stand down, he said: "No. I was hired to make United better and we've been doing that and that's what I'll continue to do."
Mr Dao was pulled off Sunday evening's flight from Chicago to Louisville, Kentucky, because the flight was fully booked, and the airline wanted to get four passengers to leave to make room for four staff members.
Mr Dao was left bloodied after law enforcement officials dragged him off the plane as he refused to leave.
The footage provoked international outrage and the Dao family issued a statement on Tuesday evening expressing gratitude for the "outpouring of support".
"This can never, will never happen again on a United Airlines flight," said Mr Munoz in Wednesday morning's television interview.
He was asked what the company would do in future if a seated passenger refused voluntarily to leave an overbooked plane based on the airline's compensation offer.
"We're not going to put a law enforcement official to take them off," he said. "To remove a booked, paid seated passenger, we can't do that."
Mr Munoz was asked if Mr Dao, who has been undergoing treatment at a Chicago hospital, was at fault.
The chief executive paused.
He said: "No. He can't be. He was a paying passenger sitting on our seat in our aircraft and no one should be treated that way. Period."
But on Monday Mr Munoz said Mr Dao's conduct meant employees were "left with no choice but to call Chicago Aviation Security Officers to assist in removing the customer from the flight".
Global outrage spread to Vietnam on Wednesday after it emerged that Mr Dao was born there, contradicting earlier reports that he was from China.
Vietnamese social media users have called for a boycott, despite there being no direct United flights to the Southeast Asian country.
"Watching this makes my blood boil, I'll never fly United Airlines," said one Facebook user, Anh Trang Khuya.
A spokesman for the Chinese foreign ministry said the government had "noticed the unfortunate incident in the United States".
Spokesperson Lu Kang added: "We sincerely hope that this incident will be properly settled."
On Wednesday, two of the aviation aviation security officers involved in removing Mr Dao from the plane were  "placed on leave", the Chicago Department of Aviation said, adding to one other that was suspended on Monday.
The agency added that their actions are "obviously not condoned by the department".
The US Department of Transportation is reviewing whether United complied with rules on overbooking.

The Barbados-born 21-year-old made his first-team debut in 2016, taking 28 wickets in seven first-class games.
He signed a first professional contract in August, but the club have now tied him to a longer deal.
"I'm happy to extend my contract. I can develop without any pressure and the aim is to continue to build on what I did last season," said Archer.
"My aim for next season is to take 70 wickets across all three formats, and if I can cement my place in the team then I think that can be a reality."
He also showed batting potential by making a career-best 73 against Essex and coach Mark Davis believes he is an "immense talent".
Davis added: "He is still a young man learning his trade but the signs are clearly there that he will be an integral part of our future success."
United Airlines' chief executive has said he will not quit amid an explosive backlash to video of a screaming man being dragged off a plane.
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