Firefighters had to smash the glass walls and use rope ladders to rescue people trapped on the first and second floors of the AMRI Hospital, in the eastern city of Kolkata.
The fire erupted in the basement, which was being used as a storage area.
Television images showed rescuers taking several patients on stretchers to another nearby hospital.
Police said 20 fire engines rushed to the scene and firefighters brought the blaze under control after several hours.
But the first three floors in the building were still filled with smoke and rescuers were searching for others who may still be trapped.
West Bengal chief minister Mamata Banerjee said most of the dead had suffocated and implied negligence caused the tragedy.