The 28-year-old suffered the injury at a training session before Sunday's first Twenty20 and will be repl7aced by fellow paceman Andrew Tye.
"Kane reported some left side pain at training prior to the opening T20I in Vizag," Australia team physiotherapist David Beakley said in a statement.
"Unfortunately he has not recovered well enough in order to play a further part in the tour.
"Kane will return home to continue his rehabilitation, and we (will) monitor his progress over the coming weeks."
Australia are already without leading fast bowlers Mitchell Starc and Josh Hazlewood for the limited-overs tour.
The touring side won the first match by three wickets at Visakhapatnam. The second and final Twenty20 will be played in Bengaluru later on Wednesday.
The two teams will then play a five-match one-day international series, starting in Hyderabad on Saturday.
(Reporting by Sudipto Ganguly; editing by Amlan Chakraborty)
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