The 50-year-old released a statement via the League Managers Association in which he spoke of his gratitude to Sir Alex Ferguson, the fans and the staff at Old Trafford.
There was no mention of the United players, who have kept tight-lipped since Moyes left the club. Even those players normally so active on Twitter have remained silent on the matter.
Moyes was sacked as manager by the club on Tuesday, and former United captain Roy Keane said some of the players "should be ashamed of themselves".
Ryan Giggs has been appointed interim manager and was assisted at training on Wednesday morning by former team-mates Paul Scholes, Phil Neville and Nicky Butt.
Moyes said he felt proud to have taken the team to the Champions League quarter-finals but that he understood fans' frustrations at the overall results.
He said in a statement: "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.
"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: "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.
"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.
"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."
It has emerged that Scholes is now back coaching at the club - Moyes tried to persuade him to join his backroom staff at the start of the season, but he failed.
"It's great to see Paul Scholes here at the Aon Training Complex today, assisting Ryan Giggs, Nicky Butt and Phil Neville," read a post on Manchester United's official Twitter feed.
Giggs, Neville, Scholes and Butt are all graduates of the famed 'Class of '92'.
The United board hope Giggs and his staff can restore some pride after a woeful season under Moyes.
Moyes' tenure was often dogged by rumours - always denied by the Scot - that he did not get on with some of his players.
One highly-placed source at United revealed the board became concerned that Moyes could not get the best out of them.
The source said: "The players didn't look up for it," during the latter stages of the former Everton manager's reign.
Although Giggs has no experience in management, he will command the total respect of his players, according to his former team-mate Quinton Fortune.
"In the last few seasons, and especially this season, I've seen him stepping in and helping out with the coaching. The amount of respect that the players have for him is unbelievable," Fortune told MUTV.
"I wish Giggsy all the best."
Giggs is expected to take charge of United's final four games but the 40-year-old is understood not to be in the running to be the club's next full-time manager.
The front runner to replace Moyes is Holland coach Louis van Gaal, who has made no secret of his desire to move to England once his contract expires after the World Cup. Real Madrid manager Carlo Ancelotti is also a candidate.