The 19-year-old posted a picture of the Manchester United goalkeeper with two gorilla emojis.
Garnacho soon deleted the social media post but the Football Association have asked for his observations via his club.
But Onana has defended his teammate and said that no further action should be taken.
The goalkeeper wrote on Instagram: "People can't choose what I should be offended by. Garnacho meant power and strength. This should go no further."
Onana later posted a picture of him hugging Garnacho during a match, demonstrating the pair are still friends.
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The social media post came late after United's win over Copenhagen in the Champions League.
Erik ten Hag's side had claimed a 1-0 win courtesy of a Harry Maguire header but Onana proved pivotal after saving a last-gasp penalty.
The Cameroon international has come under fire this season after replacing David de Gea as United's No. 1.
But it was a moment of redemption for the ex-Inter Milan star, whose true transfer fee was revealed this week.
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Garnacho took down his social media post not after uploading it but it had already been widely shared.
Despite Onana's protestations, Garnacho could fall foul of rules regarding racially sensitive social media posts.
It would not be the first time a United player has done so in recent history with Edinson Cavani being found guilty in 2020 during his time at Old Trafford.
The Uruguayan had written "Gracias negrito" in a response on social media after being congratulated for scoring twice in a 3-2 win against Southampton in November.
Cavani was made to serve a three-match ban, fined £100,000 made to complete face-to-face education around the issue.
Meanwhile, Garnacho was also accused of 'childish behaviour' during United's win on Tuesday night for his actions before the penalty was taken.