The England and Wales Cricket Board (ECB) said scans had revealed the injury, sustained as Hales attempted a catch in the outfield and jarred his hand on the ground.
The 28-year-old will return home on Saturday to see a hand surgeon.
The ECB said a replacement in the T20 squad for a three- match series starting on Jan. 26 would be announced in due course.
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India secured a series-clinching 15-run victory in the second one-dayer, with Hales dismissed for 14. The final game of the series is in Kolkata on Sunday.
(Reporting by Alan Baldwin, editing by Ed Osmond)
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