He was initially banned from all football-related activity for four months, as well as nine international matches, although that did not stop Barcelona from making the 27-year-old their marquee summer signing from Liverpool.
Although the four-month ban stands, the Court of Arbitration for Sport has subsequently relaxed the "excessive" elements of the punishment, meaning Suarez can now train and play in friendlies for his new club and country, and the former Ajax forward is now keen to move on from the incident.
"Of course I was concerned - not just about the transfer to Barcelona but more for me as a person. But, as I've said, that's in the past now and I'm trying to concentrate on Barcelona," he said at his official presentation.
"I'm here to do important things at Barca. Everything that happens in the past, that needs to be forgotten.
"If I start thinking about everything I've gone through, I wouldn't be able to sleep."
Suarez, who had previously been banned for biting PSV's Otman Bakkel and Chelsea's Branislav Ivanovic, initially denied he had bitten Italy's Chiellini in Uruguay's 1-0 win, but then issued a statement later confirming his guilt.
Asked whether he should have apologised earlier, he added: "I don't like having to go back.
"That's something that happened and afterwards I was really depressed and didn't feel like doing anything.
"But then I spoke to my psychologist and he said I had to face it and say sorry.
"I did that and now I would rather focus on the present which is Barcelona."
The forward switched Anfield for the Nou Camp earlier this summer although the FIFA ban meant he could not be unveiled until Tuesday, a day after he made his debut as a substitute in the 6-0 friendly win over Mexican side Leon.
It has been worth the wait for Suarez, though, who said: "I've dreamed of playing for Barca for years and now it's true. I never thought this moment would come.
"These are spectacular experiences, really emotional. It was a great feeling to play in the Camp Nou, I loved it."
The chance to play in a formidable attacking trio, alongside fellow South American superstars Neymar and Lionel Messi, will spur on Suarez.
"Leo and Neymar are spectacular, admirable players. It was just incredible watching them yesterday," he said.
"We all know what they can do for Barca and I'm here to add to that and get Barca back to winning titles."
Suarez held up the number nine shirt alongside Barca's sporting director Andoni Zubizerreta and vice president Jordi Mestre, whose comments over the transfer fee could make waves on Merseyside.
While the fee for Suarez had been widely reported as £75million, Mestre said: "Suarez's transfer cost 81 million euros (£65million)."
- Sports & Recreation