Garnacho used the emojis alongside an image of himself celebrating with goalkeeper Andre Onana after the summer signing from Ajax saved a last-minute penalty against FC Copenhagen.
Onana has expressed his support for his team-mate, insisting that he should face no punishment for the post, which was swiftly deleted.
Manager Erik ten Hag also defended his player and the FA has now confirmed that Garnacho will not face any sanctions.
The governing body's statement read: "We sought the player's observations as part of our investigation, and he explained that the use of two gorilla emojis was intended to highlight the strength and power of his team-mates - specifically Andre Onana and Harry Maguire - following the pivotal roles that they played in Manchester United's win over FC Copenhagen.
"We are satisfied with Alejandro Garnacho's explanation and the context that it provides, so we will not be issuing disciplinary proceedings on this occasion.
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"We also intend to liaise with Kick It Out and the PFA to discuss other potential issues around the use of emojis and similar forms of messaging, and education that can be provided about them."
There is a precedent for such issues, as both Bernardo Silva and Edinson Cavani were previously banned for social media posts that were deemed offensive.
The FA's confirmation means that Garnacho will be available to play for United against Copenhagen in the Champions League, before they take on Luton Town in the Premier League this weekend.