The 26-year-old batsman, who has been playing for Rising Pune Supergiants, is returning home for treatment, Cricket Australia has confirmed.
Australia start a tour of the West Indies for a triangular one-day series involving the hosts and South Africa at the end of May.
Smith scored his first Twenty20 century in his penultimate match for Pune.
Australia captain Steve Smith has been forced to end his Indian Premier League stay early because of a wrist injury.