Coetzee experienced discomfort in his pelvis, which got progressively worse while bowling on day three of the match on Thursday, where South Africa won by an innings and 32 runs.
He underwent scans on Friday which revealed the extent of the injury, a statement from Cricket South Africa said.
No replacement has been added to the squad.
Coetzee was part of a four-man seam attack for South Africa at Centurion where they bowled India out twice. But the second, and final, test at Newlands in Cape Town will likely see the return of spinner Keshav Maharaj to the home line-up.