Carrick served as assistant to Ole Gunnar Solskjaer before taking over as caretaker when the Norwegian was sacked.
Ralf Rangnick will take over as interim manager this weekend, and Carrick has decided to walk away following Thursday's win over Arsenal.
"It's not been an easy decision to make but I feel it's the right one," Carrick told Amazon Prime.
"I was going to take time off after I finished playing and it never happened. It feels like the right time to step away and what a way to finish.
"It's 100% my decision. Over the last week I was conscious I respected the club and the manager coming in. I thought it was the right thing to do for the club and for Ralf [Rangnick] and I'm quite happy with that.
"We were in a situation where there was a responsibility to see these games through. The loyalty to Ole is a little bit of a factor but there were a lot of things that came into my decision.
"I've had great times, great memories and I'm proud of the players over the past three games.
"I just told them and they were a bit shocked and surprised, a bit emotional in the changing room. I held it together just about. It's not been easy to keep it away from people but I had a job to do."
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