The rescuers believe the kite, weakened by the heat, fell to the ground from its nest in a tree nearly 15 metres (50 feet) above.
Hospital curator Sherwin Everett has worked at Jivdaya -- "compassionate life" in the local Gujarati language -- since 2010.
He says this year's heatwave has been among the worst for local birds he has ever seen.
Heatwaves have killed more than 6,500 people in India since 2010, but Shah and Everett are also calling on the public to also be considerate of the welfare of any wildlife they come across.
"We are expecting until July to have multiple heatwaves and the temperature will get worse," Everett said.
"Right now we have received quite a lot of birds that have been dehydrated," he added. "But the upcoming months seem to be pretty dire for us as well."