The former captain of his country's U19 side had already been banned from all international teams for life by the Greek Football Federation for making the gesture after scoring the winning goal in a 2-1 Super League victory over Veria on Saturday.
"AEK have decided to leave the player out of the squad until the end of the 2012-13 season ... thus imposing the worst possible punishment (we can)," the club said.
"Whether the player will remain at the club beyond this season will be judged in the summer after which time he will firstly have been given the opportunity to demonstrate in practice his behaviour was the result of immaturity and secondly whether or not his possible reinstatement to the squad does not cause any negative effects within the club."
On Monday, Katidis asked to be dropped from AEK's first team and called his actions "totally unacceptable".
"I feel terrible for those I upset with the stupidity of my act," he said.
"I also understand fully the reasons for the decision made by the Greek Football Federation to which I owe a huge apology as it has helped me to get where I am in the professional game."
Katidis joined AEK from Aris Salonika last year after impressing in Greece's run to the final of the European Under-19 Championship where they lost to Spain.
"I cannot turn the clock back but I want to clarify I am not a fascist or neo-Nazi or racist," he said. "I have a step brother from Puerto Rico and all my family are from the Black Sea and have experienced racism in the worst ways.
"There is no reason to look for excuses. I was completely wrong and I will pay for it with a lot of pain because despite being only 20 my career is now at risk.
"I sincerely apologise to my team mates and everyone involved with the club that I have insulted in not knowing exactly what I had done in my celebration. The fact I did not know what I was doing is no excuse," the midfielder added.
Katidis was heavily criticised by political parties and on Twitter and Facebook following the incident at the Athens Olympic Stadium.
Last Sunday marked the 70th anniversary of Greek Jew deportations in Nazi concentration camps in the Second World War.
"The player's action to salute to spectators in a Nazi manner is a severe provocation, insults all the victims of Nazi bestiality and injures the deeply pacifist and human character of the game," the EPO said.
AEK are 11th in the league with 29 points from 26 games.