Carrick, in charge for three matches, announced his decision after United’s 3-2 win over Arsenal on Thursday.
“After a lot of thought and deliberation, I have decided that now is the right time for me to leave,” said Carrick.
Ralf Rangnick has been appointed interim manager until the end of the season.
Carrick, a five-time Premier League winner during 12 years as a player at United, took over as caretaker boss after Ole Gunnar Solskjaer was sacked on 21 November.
He took charge of the 2-0 Champions League away win to Villarreal on 23 November before dropping Cristiano Ronaldo to the substitutes bench for the 1-1 draw at Chelsea last Sunday.
Carrick announced his decision to leave minutes after the Arsenal victory which lifted United into seventh spot in the table.
“I’ve just told them [the players] and they were a bit shocked and surprised, a bit emotional in the changing room,” Carrick, speaking to Amazon Prime, said.
‘Loyalty to Solskjaer was a bit of a factor’
Carrick, who also helped United win the Champions League in 2008, was first-team coach under Solskjaer.
The 40-year-old said that loyalty to his former Red Devils team-mate was a “little bit of a factor” behind his decision to leave.
“It’s not been an easy decision to make but I feel it’s the right one,” Carrick added.
“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. I’m quite happy with that.
“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.”