The 24-year-old moved back to the club he grew up with on Monday in a deal understood to be worth an initial €29 million (£25.4m).
Fabregas says he was disappointed to have been unable to speak publicly to Gunners fans to clarify his intention to move back to Barcelona, but praised the current squad and wished them well.
“I'm sorry I could not speak to them and I'm sad about leaving as I spent one third of my life there,” Fabregas told reporters at a press conference held at Camp Nou.
“It was a sad moment for me but life goes on. It was very emotional saying goodbye to my team-mates and particularly when I said farewell to Arsene.
“Jack Wilshere and Aaron Ramsey are great players and one day Jack will be England captain.
“I wish them well, especially in qualifying for the Champions League against Udinese and I thank the fans and club for the last few years.
“And I also want to thank Pep Guardiola for the effort he made and the confidence he gave me.
“I wasn't ready two or three years ago but I'm prepared now.
“But I knew I wanted to be with the best in the world. I've waited a long time for this but it was worth it. It's a special day, one of the best of my life.”
He was named William Gallas’ successor as skipper for the Gunners in 2008 but was unable to bring silverware to the side, a feat that he admits was devastating not to achieve.
“The biggest regret I will have in my career is I did not lift a trophy as Arsenal captain,” he added.
“More than just not winning trophies and medals, it was the routine fact of doing the same things over and again, trying to win and then always finding, one way or another, that we just didn't have enough to get over the finishing line.
“We just didn't seem to have what it took in the final sprint for a trophy. But Arsenal are a great club and they will push on without me just like they did when Patrick Vieira and Thierry Henry left.
“The club is always bigger than the player and I am sure they will be strong again.”
The Spain international, who already has 58 caps and six goals to his name for his country, made the move to Catalonia after a long and protracted transfer saga which began last summer.
Despite his move away from the Premier League, the midfielder confessed he will always have a deep respect for Arsenal manager and mentor Arsene Wenger.
He added: “I will never have enough words to describe what he has done for me. I will always have him in my head and my heart.
“People who ask about why I renewed my contract with Arsenal for so long need to know I renewed four years ago when I was only 20.
“[Barcelona manager] Pep Guardiola was not in charge at Barca then and there was no 'sign of life' in terms of what Barca thought about me. I was still just a kid.”