The French full-back sustained the back problem during the Gunners' Champions League quarter-final first leg at Villarreal on April 7.
Now, following the club's exit from the Champions League against Manchester United on Tuesday, Clichy's season has officially been concluded and he will return for pre-season training in July.
Arsenal mananger Arsene Wenger told the club's official website, www.arsenal.com: "It is finished [for Gael].
"We would have put a little pressure on the recovery period had we gone to the Champions League final.
"But now we will not take a gamble."