United needed neighbours City to lose their last game at Swansea on Sunday to have any hope of claiming a spot in the Champions League. But City drew 1-1 at the Liberty Stadium.
Midfielder Mata wrote in his blog for manutd.com: "We will have to close the Premier League season on Tuesday. It's going to be a bittersweet farewell to Old Trafford, now that we know we won't be playing the Champions League next season.
"It leaves us with a bad taste because, even if we enjoyed good moments together throughout the season, we weren't consistent enough to qualify for the top European competition.
"It's true that we found many difficulties along the way, especially if we talk about injuries (up to 15 players in several weeks, as far as I remember), but Manchester United must be ambitious and always fight for everything."