The incident happened shortly after Arsenal had been awarded a penalty which was converted by Bukayo Saka to hand Mikel Arteta's side a 3-2 win at the Emirates Stadium.
The FA were investigating a comment alleged to have been made by one of the players in the heat of the moment.
However in a statement released on Wednesday, the FA have announced that investigations into the incident have been closed without any charge being issued to either player.
The statement read: "Whilst The FA is entirely satisfied that the allegation was made in good faith, it is equally satisfied that there is no case to answer."
It explained: "The investigation included taking witness statements from the complainant and accused, along with an additional six players who were within proximity of the alleged incident, reviewing multiple angles of video footage, and seeking independent linguistics experts evidence.
"None of the witnesses heard the alleged comment, and the player accused strenuously denied the allegation throughout."
The statement added: "The FA continues to take all allegations received extremely seriously and would encourage anyone who believes that they have been the subject of or witness to abuse to report this through the appropriate channels.
"The FA receives a number of complaints in any one season and thoroughly investigates each one that it receives. Following consideration of all of the evidence received, not all cases will result in a charge being issued."
