NEW DELHI—When top American naval engineers recently inspected India’s first locally made aircraft carrier they expected to find a near battle-ready ship set to help counter China’s growing sway in the Indian Ocean.

Instead, they discovered the carrier wouldn’t be operational for up to a decade and other shortcomings: no small missile system to defend itself, a limited ability to launch sorties and no defined strategy for how to use the ship in combat. The findings alarmed U.S. officials ​hoping to enlist India as a bulwark...