The 18-year-old was shown a straight red card away to Barnsley back in October for calling an opponent "gay boy" after he had the opened the scoring just seven minutes into the match.
In a statement, Fletcher expressed remorse for his conduct.
"I am truly sorry for the offensive word that I used in the heat of the moment," he said.
"Despite the fact that I had no intention to use the term as a homophobic insult, I completely understand that such language is unacceptable and immediately apologised after the game.”
As reported by the BBC, Fletcher was also fined £1,500 and must attend a mandatory face-to-face education programme or face further suspension until it is completed.
"I want to be clear that this momentary lapse of character absolutely does not reflect my beliefs or values," Fletcher added.
The teenager made his first-team breakthrough at Old Trafford this season under former manager Ruben Amorim, with three Premier League appearances to his name at the end of December.
Reports state United have taken steps to enhance players' understanding of discriminatory language and its harmful effects.