The England midfielder ended 12 years at Villa Park when agreeing a £12million transfer to Premier League rivals City at the beginning of June.
Barry's switch to City, who finished 10th in 2008/09, has caused controversy after he stated last summer that he wanted to play Champions League football when requesting a transfer, sparking a long-running, and ultimately unsuccessful, saga involving Liverpool.
The 28-year-old's decision to join City has, therefore, earned condemnation from some quarters as Mark Hughes' squad will not be playing European football next season.
However, former Villa captain Barry, who wrote an open letter to former fans when leaving the club, insists he does not see a reason to say sorry for his move to Manchester.
In a press conference shown live on Sky Sports News, he said: "People made out that I was apologising for it. I don't feel the need to apologise for leaving Aston Villa.
"I just wanted to state my reasons for leaving. I had been there for 12 years and had built a good relationship with the fans.
"Obviously it went a bit sour at the end of last summer for the way that saga ended."
Having completed his move, Barry is hoping to help money-rich City become an established force in the top flight.
Barry admits that after holding discussions with Hughes, who is funded by billionaire owner Sheikh Mansour and is expected to be busy in the summer market, he has been impressed by City's aspirations.
The 30-time England cap is sure that he has made the right move after avoiding the drawn-out speculation which surrounded his future 12 months ago, and confesses that the speed with which City conducted business played a part.
He added: "The club (Villa) have put the price on me and City have come in and paid the money really early in the transfer market.
"Once I talked to the manager, and the ambition about where they think they can head, and I spent a bit of time thinking about it and talking to a few friends and family, I made my decision and I'm very confident I've made the right move.
"Certainly from last summer, it was a tough summer for me, it dragged on and on.
"When a club are willing to pay the money early on and are willing to get a deal done - I felt very wanted."
He added: "We have already seen the type of player Manchester City have been able to attract with the likes of Robinho and the type of players that they are being linked with.
"I think that this club has got everything in place to go and achieve big things and challenge right at the top.
"I am really looking forward to being a part of that and turning Manchester City into a team challenging for big honours. That would be a big thing for me."
Barry is also adamant that he is not thinking about what could have been after his failed move to Anfield in 2008.
He said: "That is completely out of my thinking, worrying about Liverpool or what could have been. That is all in the past and something you can't do anything about.
"That all stemmed back to last year. I stated I wanted to play Champions League football and Liverpool showed a lot of interest in me. A deal couldn't be reached, that is all in the past now."