"I would like to publicly apologise to you, Chris," he said in a statement. "I was out of line and I was wrong."
It comes after the Oscars film academy condemned Smith over the incident and announced a formal review.
The star slapped Rock in the face on stage after the comic made a joke about the actor's wife, Jada Pinkett Smith.
Rock had taken aim at Pinkett Smith's shaved head, a result of the hair-loss condition alopecia.
Moments later Smith picked up the first Oscar of his career for playing the father of tennis legends Venus and Serena Williams in King Richard.
"Violence in all of its forms is poisonous and destructive," Smith said in the statement, which was posted to Instagram.
"My behaviour at last night's Academy Awards was unacceptable and inexcusable. Jokes at my expense are a part of the job, but a joke about Jada's medical condition was too much for me to bear and I reacted emotionally."
Smith apologised directly to Rock, stating he was "out of line". He also apologised to the Academy and the Williams family.
"I deeply regret that my behaviour has stained what has been an otherwise gorgeous journey for all of us," he wrote.
Prior to Smith's apology, the organisation behind the Oscars said in a statement that it "condemns the actions" of Smith.
"We have officially started a formal review around the incident and will explore further action and consequences in accordance with our Bylaws, Standards of Conduct and California law," it said.
Will Smith and his PR team will be hoping his apology will be enough to salvage his career and his relationship with the Academy, and douse the blaze of publicity surrounding Sunday night's altercation.
His statement has been written with extreme care. Smith is careful with his language, explaining why he reacted the way he did, without trying to excuse it.
But his apology does not mean this story is over. Apart from anything else, Rock will eventually have to comment publicly too. The signs so far suggest he intends to shrug the episode off with good humour.
The comedian declined to file a police report, and it was reported that he joked backstage after the punch: "That's the only time you get hit by Muhammad Ali and it doesn't leave a scratch," referring to Smith's role in the boxer's biopic.
But even when the headlines about the punch eventually start to fade, Sunday's events will follow those involved for years to come. It would be impossible for any journalist or chat show host to interview either of them while they're plugging a new film without bringing it up.
Meanwhile, the Academy's Board of Governors is reportedly set to meet on Wednesday to discuss whether or not any further action should be taken against Smith.
Some had suggested that his Oscar could be revoked, but that is unlikely. He could still receive a gentler sanction such as having his Academy membership suspended, but that may now be less likely following his apology.
Only one thing is certain - worldwide appetite for this story remains high, and fans will be interested to see where it goes next.
Rock's joke referred to the 1997 film GI Jane, in which Demi Moore played the title role with a severe buzzcut.
Pinkett Smith rolled her eyes at the comment, while Smith appeared to initially laugh and clap his hands before he was seen on stage, walking up to Rock.
The comedian looked stunned in the immediate aftermath of the incident, but told the audience: "That was the greatest night in the history of television."
The Los Angeles Police Department later told Variety that Rock had "declined to file a police report" following the event.
After the incident, Rock handed over the best documentary prize, which was the reason he was on stage.
He is yet to publicly comment further on what happened.
Accepting the Oscar for best actor for King Richard soon after, Smith said Richard Williams was a "fierce defender of his family" and that he himself was "being called on" in his life "to love people and to protect people".
"Art imitates life," he said. "I look like the crazy father, just like they said about Richard Williams. But love will make you do crazy things."
He ended it saying he hoped the Academy would invite him back.