Nicolas Pepe says he is "deeply sorry" for his sending off in Arsenal's 0-0 draw against Leeds on Sunday.
Pepe headbutted Alioski in the second half of Sunday's match - which ended 0-0 - leaving referee Antony Taylor no chance but to brandish a red card following a VAR review.
Arsenal boss Mikel Arteta was unhappy with the Ivorian, the club's record signing, after the game and said his actions were "unacceptable".
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Pepe apologised on Instagram, saying: "Yesterday I have let my team down at a crucial time of the game and there is no excuses for my behaviour. I am deeply sorry and would like to apologise to the fans, my team-mates, my coach and everyone else at the club."
Sky Sports pundit Patrice Evra said he was "disappointed" by Pepe's actions but felt Alioski was looking to get the Ivory Coast international sent off.
Both Leeds United and Arsenal condemned "vile" abuse aimed at Alioski and Pepe on social media after Sunday's incident.
A statement from Leeds read: "The vile abuse directed at Ezgjan Alioski and Nicolas Pepe on social media after yesterday's Premier League game with Arsenal will not be tolerated by Leeds United.
"We will work closely with the police and the footballing authorities to ensure those responsible are identified and punished."
Arsenal added: "We utterly condemn the vile abuse directed at Nicolas Pepe and Ezgjan Alioski on social media following our match against Leeds. This is completely unacceptable and we will be working with the police and authorities to do everything we can to trace and prosecute the culprits."
Sky Sports' Peter Smith:
Mikel Arteta has a creativity crisis on his hands at Arsenal.
It is now seven hours and 56 minutes since his side scored from open play in the Premier League - and a formation change and shift in position for Pierre-Emerick Aubameyang did little to solve the problem in Sunday's 0-0 draw at Leeds United.