A woman must only treat other women and men must treat other men!

Only if there is no one to treat the opposite gender that this might be permissible but in the presence of someone else in the room so that there wouldn’t be any seclusion.

I am a doctor and I have both male and female patients, I have to take their pulses, blood pressure , examine their chest heart beat via stethoscope and other aspects of medicine, usually I donot see the private parts of the patient. But still in order to examine them I have to touch their hands or legs or chest.. Is my income halal if I am touching a non Mahram for it?