The Middlesex paceman did not take part in training on Tuesday having sustained the problem during his side's warm-up match against India A.
The first Test gets underway in Ahmedabad on Thursday, and captain Alastair Cook has confirmed that his teammate won't be back in time for the match.
"Steven will not be available for tomorrow's Test", said Cook. "It would be too big a risk."
Tim Bresnan is expected to be named as Finn's replacement.