The 44-year-old former midfielder arrived at United's training ground on Tuesday, and it was later confirmed that he had signed a contract that will last until the end of the season.
"There is still a lot to fight for this season. We are ready to pull together and give the fans the performances their loyal support deserves," said Carrick.
Former England assistant manager Steve Holland will also join the coaching staff, which includes Jonathan Woodgate, Travis Binnion, Jonny Evans and Craig Mawson.
Carrick previously served as an assistant coach at Manchester United and was in charge briefly after Ole Gunnar Solskjær was fired in 2021. The Norwegian has been interviewed to succeed Amorim, but United have now instead chosen his assistant as the new coach.
Carrick's only long-term tenure as head coach was for Middlesbrough, 2022–2025.
Manchester United are currently seventh in the Premier League and were knocked out of the FA Cup this weekend.
