Kumar was admitted to Nalanda Medical College and Hospital in the state capital Patna on Thursday with a gunshot wound in his abdomen and died the next day.
The family say they were with Kumar's body after his death but left late in the night and found the eye missing upon their return.
"A medical team has been constituted to probe the matter. It's a serious issue and strict action must be taken against those found guilty," the hospital's medical superintendent Binod Kumar Singh said.
"A group of doctors suspect that rats might have gnawed the eye. All aspects pertaining to the incident are being probed. We are also waiting for the autopsy report. How the eye was gouged out can only be known after we get the autopsy report. The hospital administration has also lodged a formal complaint with the Alamganj police station in this regard."
Police say they have registered a complaint.
They have also examined surveillance footage from the hospital, but haven't found anything related to the matter yet, Rajiv Kumar, a local police officer, said.
NDTV reported a family member of Kumar alleging they believe the eye was taken by hospital staff who may either be involved in the trafficking of body parts or associated with the people who shot him.