Carrick was relieved of his duties on Thursday following a disappointing start to the campaign, with United sitting 18th in the Premier League after collecting just one league victory in their opening 10 matches.
The former midfielder was appointed permanently after guiding United to third place last season, having initially taken charge following the departure of Ruben Amorim. But the team failed to maintain that momentum, and the club decided to make another change.
Following the confirmation of Mauricio Pochettino as his successor, Carrick published a message on Instagram thanking the players, staff and supporters.
"I've had the immense privilege of leading Manchester United, the football club that I love. There are very few who get to live and feel it, and for that I'm incredibly grateful," he wrote.
"With that comes huge responsibility, and to be at Manchester United, you need to understand exactly what that means.
"It's different to anywhere else. Ultimately, you need to push to be the best, but there is a culture, there are values and there are behaviours that you have to live by and respect, which have been built over a long period of time."
Carrick reserved particular praise for the supporters, acknowledging their continued backing despite the team's struggles.
"I want to say a big thank you to the supporters. I must say, you are a special, rare breed.
"We've had some great times this year and moments we'll all remember. The best thing about being in my position was seeing so many of you lose yourselves in the emotion of those moments, leaving the stadium desperate to come back again next time.
"I just wanted to give you something back. That love never dies. I know we've not had the results lately, but you've shown me nothing but love and support in times when the easiest thing, and what most end up doing, would be to turn the other way.
"That means more than a win or two. That's what makes you who you are. That's why I fell in love with this club all those years ago."
He also expressed gratitude to the coaching staff and the wider group of employees who work behind the scenes at Old Trafford and Carrington.
"A special thank you to my coaching staff. You gave me everything you had, every single day, and I couldn't have asked for any more. I'm incredibly grateful for your loyalty, hard work and support throughout.
"To all the staff behind the scenes, thank you for everything you do for this football club. So much of your work goes unseen, but it never goes unappreciated."
Carrick also addressed the players, wishing them well as they begin working under a new head coach.
"And to the players, thank you for your efforts and the moments we've shared. I wish you all the very best for what's ahead. I want nothing but success for Manchester United."
The former England international's departure brings an end to another short managerial spell at the club he represented as a player. Despite the disappointment of losing his position, his closing message made clear that he intends to remain a supporter.
"This club has been such a big part of my life and that will never change. I'll always be proud to have played my part and I'll always want the very best for it.
"I'll be right there with you from now on, watching from the side-lines, cheering the lads on. Michael."