Suarez issued the apology in English and Spanish on his Twitter account for the incident that led to his four-month ban from football.
The Liverpool and Uruguay forward appeared to bite Italy centre-back Chiellini in the Group D fixture on June 24, the third incident of its kind involving Suarez, but initially denied the accusation and told FIFA that he lost his balance and fell on to his opponent.
He has now been back in Uruguay for a few days and has said sorry to Chilleini, whilst also stressing that he will never be caught up in such controversy again.
His statement said: "After several days of being at home with my family, I have had the opportunity to regain my calm and reflect on the reality of what occurred during the Italy-Uruguay match on 24 June 2014.
"Independent from the fallout and the contradicting declarations that have surfaced during these past days, all of which have been without the intention of interfering with the good performance of my national team, the truth is that my colleague Giorgio Chiellini suffered the physical result of a bite in the collision he suffered with me.
"For this: I deeply regret what occurred. I apologise to Giorgio Chiellini and the entire football family. I vow to the public that there will never again be another incident like this involving me."
Chiellini responded on Twitter within minutes, accepting the apology and calling on FIFA to show leniency to Suarez.
"It's all forgotten," the Italian wrote on Twitter. "I hope FIFA will reduce your suspension."
Uruguayan president Jose Mujica condemned the FIFA decision to ban Suarez from all football activity for four months as "Fascist" whilst other sporting professionals have called for the Liverpool star to have medical intervention and counselling.
Suarez has bitten two other players - Otman Bakkal of PSV Eindhoven and Branislav Ivanovic of Chelsea - in previous matches.
Former Liverpool manager Kenny Dalglish is one of those to have stated that Suarez needs support, and he does not believe the Reds will 'turn their back' on the player.
The Scot said in his Daily Mirror column this week: "It is my belief that when you bring a player to a football club, he becomes your responsibility. You don't just turn you back on a player because he has done something wrong.
"I think you will find that Liverpool will not turn their back on Suarez, whatever the ban FIFA have decided he must serve. Of course it will be a heavy blow if the club has to do without him for the opening months of the season."
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