I hurt my hand in gymnastics, and my hand hurts under my pinky finger, it hurts to move and touch, it is a little swollen but it is not bruised?

It could be broken: I personally have had broken hand bones and didn't know it until later when i had an x-ray. It may be a good idea to see a doctor for an x-ray, just to be sure.

Hand surgeon: There are several issues that could cause your symptoms. The best care would be to have a hand surgeon evaluate you with additional imaging such as an x-ray. Continuing gymnastics without an evaluation could lead to further injury.