The left-armer took five wickets in the first two Tests, with the four-match series tied at 1-1.
The 27-year-old will return to Australia for treatment and rehabilitation.
He has been replaced by pace bowler Pat Cummins, 23, ahead of the third Test which begins on Thursday in Ranchi.
Australia fast bowler Mitchell Starc has been ruled out of the final two Tests in India after suffering a stress fracture to his right foot.