“I completely understand such language is unacceptable and immediately apologised after the game.”
Fletcher said 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.
In its written reasons, the FA said Fletcher made his comment to an opponent who had made remarks about him and his family throughout the match.
Fletcher’s twin brother, Tyler, was also playing, while Darren was in the crowd.
Fletcher told the FA he had been thrown to the floor and his Achilles had been stamped on by the same opponent earlier in the game.
United said they have been working with Fletcher to “strengthen his understanding of discriminatory language and why it is harmful”.
They said he will continue to take part in club diversity programmes in addition to the training he has been told to do by the FA.
United’s official LGBTQ+ fans group Rainbow Devils said it welcomed Fletcher’s apology and that “homophobic language has no place in football or society regardless of intention or connotation”.
“Words matter and words hurt,” it said.