Costa, an injury doubt before the first leg, limped off in the first half of the match after aggravating the problem.
Atletico said on Wednesday that they were waiting to see how the injury healed after scans found no tear in the muscle, but Costa has reportedly ruled himself out of the return leg.
"Diego Costa has suffered a muscle injury in his right thigh, with no tear," Atletico said on Twitter. "It is pending evolution."
However, AS quoted the player as saying, "I can not recover (in time)."