The Hellenic Football Federation announced the punishment in a statement on Sunday, saying the action was "a profound insult to all victims of Nazi atrocities" and they condemned it "unequivocally and categorically".
The Federation said the ban was a unanimous decision from the executive committee at an extraordinary meeting.
AEK have demanded an apology from the player and said the board would meet to decide his future.
Katidis raised his right arm in an apparent Nazi salute after scoring the winning goal in his side's 2-1 Greek Super League win over Veria.
The 20-year-old had been heavily criticised on social media for the celebration, although he has insisted he did not know the meaning of the gesture.
He reportedly claimed he was pointing to team-mate Michalis Pavlis, who is recovering from injury, in the crowd.
Katidis wrote on Twitter: "I am not racist in any way.
"I abhor fascism. I would not have done it if I knew that it meant something. I know the consequences and would not do it ever."
Katidis has played for Greece's Under-17, U19 and U21 teams, but has not been capped at senior level.