The England Under-19 international, son of United legend Darren Fletcher, was shown a red card in the 62nd minute of a
5-2 EFL Trophy defeat by Barnsley in October.
At the time it was not clear what had prompted the decision.
However, the Football Association has now confirmed that Fletcher, 18, was sent off for calling an opponent "gay boy" and that he has admitted the charge.
"I am truly sorry for the offensive word I used in the heat of the moment," said Fletcher.
"I completely understand such language is unacceptable and immediately apologised after the game, a momentary lapse of character absolutely does not reflect my beliefs or values".
He was also fined £1,500 and must attend a mandatory face-to-face education programme or face further suspension until it is completed.