Broad will return to the UK immediately to begin a period of rest and rehabilitation.
James Harris, who is currently in Pune with the England Performance Programme, will replace Broad in the squad.
The 22-year old is uncapped at international level.
Eoin Morgan will captain the side in Broad's absence with the first match taking place in Pune on December 20.
The second game will be held in Mumbai on December 22.
Pacer Steven Finn will also return to the UK after a strained disc in his lower back ruled him out of the final Test here earlier today.
Finn is expected to be available for the ODI series against India beginning in January.