NEW DELHI — Patients, some with severe eye trauma, have overrun hospitals in the Indian-administered part of Kashmir, doctors said on Wednesday, after security forces used pellet guns to break up demonstrations that have shaken the region.

Widespread protests broke out on Saturday across the Kashmir Valley over the death of a young separatist militant, Burhan Muzaffar Wani, who was killed in a gun battle with the security forces the day before. Mr. Wani had gained a following over the years, in part through his prominence on social media.

Both India and Pakistan claim the Kashmir region.

More than 30 people have been killed during the demonstrations, including one police officer, and more than 2,000 injured, said Asgar Hassan Samoon, a senior administrative officer of Kashmir from Srinagar, the summer capital of Jammu and Kashmir State.

The police also fired on demonstrators with bullets in addition to the less-lethal pellets.

As the protests faded, hospitals have been overwhelmed with people seeking treatment for eye injuries, and the Jammu and Kashmir chief minister, Mehbooba Mufti, has asked the health minister in New Delhi to send a team of specialists to Kashmir.