The bowler hurt his knee twice while fielding during England's first test victory in Hyderabad where he picked up two wickets and missed the second match in Visakhapatnam, which the hosts won on Monday to level the series at 1-1.
The ECB said it will not call up a replacement and Leach will head home from Abu Dhabi, where the team have been staying ahead of the third match in Rajkot which starts on Thursday.
"Leach will work closely with the England and Somerset medical teams in respect of his rehabilitation," the cricket board added in a statement.
England handed Shoaib Bashir his test debut in place of Leach in the second test. Bashir missed the opener following a visa delay that was later fixed.
The third game will be followed by matches at Ranchi and Dharamsala.
India batsman Virat Kohli was also ruled out of the final three tests due to personal reasons.
