The Portuguese intends to reorganise his coaching staff and will not have an immediate deputy, although he hopes Carrick could grow into such a role.
"For a victor like Michael, to win in his final match has to mean something", Mourinho told reporters.
"They finished second and could win the FA Cup but let's face it... this is one of the most boring United teams", Redknapp told the Daily Mail.
"I need to thank Jose for giving me the opportunity", said Carrick in an address to supporters after the game.
Manchester United treated their stalwart Michael Carrick to a memorable farewell on Sunday after a first-half goal by Marcus Rashford gave them a 1-0 home win over Watford on the final day of the season.
The Englishman enjoyed a positive run of games in December, leading to Jose Mourinho branding him as one of the best left-backs in Europe, but has since only made four starts.
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"Of course I have to make the decisions but my assistants have an opinion and he has".
Manchester United announced on the weekend that the 42-year-old, who spent 17 years as Jose Mourinho's assistant, will leave the club this summer, with Mourinho himself adding that his friend is "ready for a successful career as a manager". "I've been at this club, and it is such a great club, so I obviously know the club and the manager". It just fits. Hopefully it still fits in a couple of years' time.
"The final chance to cut the cloth clean. He'll teach me some things and we're going to have some success in the near future, I'm certain of that".
Carrick has been one of the experienced members of United's midfield after Scholes' retirement and now fans are concerned if his absence would create a big gap in the team. I'm not getting carried away here.
Carrick just played his final, and 464th in total, game at Old Trafford this past Sunday.
"I have a lot of memories to cherish, some I'll never forget, especially of this place".