In its written reasons, the FA said Fletcher had made comments about an opponent who had made comments about him and his family during the match.
Fletcher’s twin brother, Tyler, also played, and Darren was in the audience.
Fletcher, who made his Premier League debut against Aston Villa in December and has appeared in three games for United’s first team, told the FA that he was thrown to the floor and trampled on his Achilles tendon by the same opponent early in the match.
United said it had been working with Mr Fletcher to “strengthen our understanding of discriminatory language and why it is harmful”.
In addition to the training ordered by the FA, he will continue to participate in the club’s diversity program.
United’s official LGBTQ+ fan group Rainbow Devils welcomed Mr Fletcher’s apology, saying: “Homophobic language has no place in football or society, regardless of intent or connotation.”
It says, “Words matter and words hurt.”