The FA Premier League has rejected Tottenham's request to replay Sunday's match at West Ham.
Carrick was one of the Tottenham players struck by illness
Spurs chairman Daniel Levy wrote to the Premier League requesting the fixture be replayed after 10 players were taken ill before the match.
A statement to Tottenham said: "We find no grounds for acceding to your request for a replay."
The 2-1 defeat ended Tottenham's Champions League hopes as Arsenal took fourth place in the league instead.
The statement added: "Tottenham did have the option of not fulfilling the fixture and will have made its own assessment of the risks associated with that decision.
"It would have been for an independent commission to have decided the merits of your case, rule on any sanction and/or the appropriateness of a replay.
Spurs chairman Daniel Levy insisted the club had the support of a number of other Premier League clubs and criticised the state of the rules over such matters.
"We are disappointed, but not surprised," he said. "We continue to feel justifiably aggrieved at the course of events and remain convinced that the game should never have been played.
"If the loss of over half a selected squad to an inexplicable illness in highly suspect circumstances does not warrant the postponement of a match, then I can only despair at the state of our governing rules.
"In going ahead with the match, I chose the only safe option that would guarantee our already hard fought for place in European competition and in making this request to the Premier League, we have fought to right an injustice.
"I should stress that our purpose in taking this request to the Premier League was two-fold; firstly to seek the replay we feel we rightly deserved, but, equally importantly, to highlight these wholly inadequate and ambiguous rules that fail to regulate such incidents."
But the Premier League have now scuppered any hopes of Spurs replaying the fixture at a later date and the North London club must settle for a place in next season's Uefa Cup.
"We remain very sympathetic to the unenviable position you found yourselves in and hope that neither Tottenham or any other club will suffer the same fate again.
"However, we believe we expedited our responsibilities effectively and with due care and consideration based on the facts available to us.
"We of course realise that for Tottenham, a feeling of unfairness at the ill-fated events of last Sunday will linger, but trust that you will put this behind you and concentrate on domestic and Uefa Cup success next season."