Wiese suffered the injury to his right hand while trying to take a catch off his own bowling during a warm-up match and required surgery, which will keep him out for up to six weeks.
Wiese's injury is a blow to South Africa.
Faf du Plessis, the T20 captain, said, "David is a fantastic player for us in the T20 format. He has done really well. His bowling is going to be missed the most. He has been a banker from a captain's point of view. You know what you are going to get from him. He is pretty clever in the middle, he is a very smart bowler.
Rilee Rossouw has also been ruled out of the entire limited-overs leg of the India tour as he continues his recovery from a stress fracture to his foot he sustained against New Zealand.