The son of former United midfielder Darren Fletcher was playing for the club's Under-21 side against Barnsley in the EFL Trophy in October when he allegedly called an opponent "gay boy".
The FA charged Fletcher with misconduct, adding that the offence constituted an "aggravated breach". The 18-year-old admitted the charge and was handed a six-match suspension plus a 1,500 pounds ($2,000) fine.
"I am truly sorry for the offensive word that I used in the heat of the moment," Fletcher said in a statement.
"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.
"I want to be clear that this momentary lapse of character absolutely does not reflect my beliefs or values."
Fletcher will continue participating in Academy diversity and inclusion programmes and will undergo educational training through the FA.
Fletcher, who made his Premier League debut for United in a 2-1 defeat by Aston Villa in December, has made three substitute appearances for the first team this season. His twin brother Tyler also plays for the Old Trafford club.
Darren Fletcher was appointed United's interim manager following the sacking of Ruben Amorim in January and led them in two matches before returning to his role as the club's Under-18 coach.