The Englishman enjoyed glittering success at United, winning everything in sight under Sir Alex Ferguson.
But Carrick says he still regrets how he played against Barcelona in the 2009 Champions League final, which United lost.
"Being us, we celebrated that night [after winning the Champions League in 2008], flew back the next day and that was it, onto the next challenge," he told the club's official website.
"We won the league again the next year, won the Club World Cup, won the League Cup, almost won the Champions League again.
"We should have beaten Barcelona in Rome – that's the one that got away, for me – but that doesn't change what a special group of players that we had.
"We were absolutely relentless."