Moyes, who was sacked by United on Tuesday, said he felt proud to have taken the team to the Champions League quarter-finals, and that his time at Old Trafford will only help his future career.
He also said that he understood fans' unhappiness with the job he had done: "I both understand and share their frustration," he said in his statement.
"I have always believed that a manager never stops learning during his career, and I know I will take invaluable experience from my time as United's manager.!
He spoke warmly of the United staff but, perhaps pointedly, not the players.
"I remain proud to have led the team to the quarter-finals of this year's Champions League and I remain grateful to Sir Alex Ferguson for believing in my ability and giving me the chance to manage Manchester United," he added.
The League Managers' Association has blasted the "unprofessional manner" in which Moyes was sacked.
LMA chief executive Richard Bevan said: "The LMA is very disappointed with the nature of David's departure from Manchester United and to read extensive reports in the press, confirming David's sacking, before David himself had been spoken to officially by the club.
"Throughout his time at United, David, as he always does, has conducted himself with integrity and professionalism, values that he believes in and that have been strongly associated with the club and its rich tradition.
"It is therefore sad to see the end of David's tenure at United being handled in an unprofessional manner."
David Moyes statement in full:
"To have been appointed as manager of Manchester United, one of the biggest football clubs in the world, was and remains something of which I will always be incredibly proud," Moyes said.
"Taking charge after such a long period of continuous stability and success at the club was inevitably going to be a significant challenge, but it was one which I relished and never had a second thought about taking on.
"The scale of the manager's job at United is immense, but I have never stepped away from hard work and the same applies to my coaching staff. I thank them for their dedication and loyalty throughout the last season.
"We were fully focused and committed to the process of the fundamental rebuilding that is required for the senior squad. This had to be achieved whilst delivering positive results in the Barclays Premier League and the Champions League. However, during this period of transition, performances and results have not been what Manchester United and its fans are used to or expect, and I both understand and share their frustration.
"In my short time at the club I have learnt what special places Old Trafford and Carrington are. I would like to thank the United staff for making me feel so welcome and part of the United family from my first day. And of course thank you to those fans who have supported me throughout the season. I wish you and the club all the best for the future.
"I have always believed that a manager never stops learning during his career and I know I will take invaluable experience from my time as United's manager. I remain proud to have led the team to the quarter finals of this year's Champions League and I remain grateful to Sir Alex Ferguson for believing in my ability and giving me the chance to manage Manchester United."
- Sports & Recreation
- Manchester United
- David Moyes
- Sir Alex Ferguson