SRINAGAR, India—Mudasir Ahmed Parry says more patients have died under his watch in the past three weeks than in the entire year.

His hospital, the largest orthopedic facility here, has run short of lifesaving drugs and surgical implants due to a military clampdown by Indian authorities that has severed communications and isolated the region. It also lacks the ability to contact specialists at other hospitals, delaying critical treatment, hospital staff says.

“So...