Garnacho made the post after Onana saved a last-minute penalty against FC Copenhagen in the UEFA Champions League a fortnight ago.
The use of the gorilla emojis was deemed racially sensitive but Garnacho insisted he meant 'power and strength', and was backed by Onana.
When reports emerged that the FA were looking into Garnacho's post, Onana responded with a social media post that read: "People cannot choose what I should be offended by. I know exactly what [Garnacho] meant: power and strength.
"This matter should go no further."
The FA have now concluded their investigation and Garnacho will face no punishment.
An FA spokesperson said: "We are satisfied with Alejandro Garnacho's explanation and the context it provides, so we will not be issuing disciplinary proceedings on this occasion.
"However, we have reminded the player of his responsibilities around social media posts and the use of emojis in particular which can be interpreted in different ways.
"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."
Former Manchester United striker Edinson Cavani and Manchester City midfielder Bernardo Silva have received bans and fines from the FA for racially sensitive social media posts in the past.
