The 33-year-old is convinced that the club will go on to win silverware in the near future and says the decision to leave Louis van Gaal's side was the hardest of his career.
The 33-year-old completed his 1.9 million-euro switch to the Italian giant Monday and is adamant that United will get back to winning trophies after a difficult 2013-14 campaign.
He wrote in a statement posted on the club's official website: "After a great deal of thought I have decided the time is right for me to leave Manchester United. It is the biggest decision of my career as this club is, and will always remain, in my heart.
"Joining and playing for Manchester United has been a privilege, a dream come true and my time here has been so memorable. Through the good times and the bad times we have all stuck together and remained loyal. I have always said you don’t just join Manchester United, you join the family and long may that tradition continue.
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"This club is steeped in history and I feel privileged to have become a part of that. Over the past nine years I have met and worked with some incredible people, many of whom will remain friends for life. I would like to thank everyone at the club — teammates, coaches and in particular all the staff who made every single day in Carrington or at Old Trafford a special day.
"An immense thank you to Sir Alex Ferguson for making it all possible, for giving me the privilege to be a captain, to be inspired by the legend of Manchester United and to understand that nobody is bigger than the club.
"My biggest thanks go to the fans. We have had some fantastic times together, we lifted all of those trophies together and that night in Moscow will live with us all forever. Every single time I stepped out in that shirt I knew I was playing for the badge and for a very special club.
"My passion for this club will live on and I know you guys will go on to enjoy more success in the coming seasons."