The 33-year-old Carrick, who joined United from Tottenham Hotspur in 2006, takes over as Wayne Rooney’s deputy from Fletcher who was handed the role at the start of the season.
Scottish midfielder Fletcher has struggled for first-team opportunities, playing just 49 minutes since the end of August.
“I have three captains,” manager Louis van Gaal said. “You have a sequence: Wayne Rooney is one, Carrick is two, Fletcher three.
“He brings a lot of experience but also a lot of composure. He can transfer my philosophy on to the pitch.”
Third-placed United travel to Aston Villa in the Premier League on Saturday.