The Serbian centre-back leaves United at the end of the season, bringing an outstanding eight-and-a-half year stay to an end, and he made one last appearance at Old Trafford against the Tigers after replacing the injured Phil Jones on 22 minutes, following it with a heartfelt speech on the pitch after the final whistle.
Vidic helped United to five Premier League titles, three league cups and the Champions League and, in Carrick's opinion, deserves to be remembered as one of the club's greats.
"You want to win the game but, at the same time, you've got to celebrate Vida," Carrick told MUTV. "We have to celebrate the service he's given the club for so long, having achieved so much.
"He's a leader and one of the best defenders the club has had. It's sad that he's leaving because he'll be missed as he's a top player and an important part of the squad. Sometimes you've got to move on and we respect that.
"I'm sure he must have been touched by the reception he got from the crowd. He's a man of few words so I was a bit surprised to see him get on the microphone at the end. You could see he was touched and there was definitely some emotion there. He's got one game left but that was his last one at this special place. He'll be missed."