The Ivory Coast forward was sent off in the 51st minute by referee Anthony Taylor following a VAR review of a clash between the two players, which revealed Pepe thrusting his head forwards and into Alioski’s face in an off-the-ball clash.
Arsenal manager Mikel Arteta labelled the incident “unacceptable” a number of times after the match, and confirmed that he had spoken to Pepe in the Arsenal changing room immediately after the Premier League draw that left the Gunners down in 12th in the table.
Pepe issued an apology on his Instagram page that was quickly shared by the Arsenal Twitter account, with the 25-year-old admitting he had no defence for his actions.
"I let my team down at a crucial time of the game and there are no excuses for my behaviour," Pep said on Instagram.
"I am deeply sorry and would like to apologise to the fans, my teammates, my coach and everyone else at the club."
Arsenal and Leeds issued statements following the incident to condemn “vile” racist abuse of both players on social media, and vowed to work with police to trace those responsible and take any appropriate action.
An Arsenal club statement read: “We utterly condemn the vile abuse directed at Nicolas Pepe and Ezgjan Alioski on social media following our match against Leeds United.
“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.”
Leeds United said: “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.”
Pepe will miss Arsenal's next three Premier League matches against Wolves, Tottenham and Burnley, but could feature in Thursday's Europa League game at Molde.