Some of the majors just require you to hold a 1st class medical (SWA & Fedex for sure). I think that AAL and DAL used to give you an astronaut physcial with their doctors but what they'll be doing when they start hiring again is anyone's guess (since they'll probably have computerized, gotten a new chief of pilot hiring, revamped the app and testing, etc). There were some court cases associated with the American's with Disablities Act. UAL went to the Supreme Court over their vision requirement (used to be 20/400 correctable I think, might have been 20/200) and they won, but then they dropped the requirement anyway. Seems like the airlines are tending towards the view that if it is good enough for the Gov't/FAA, i.e. you can hold a 1st class, then it is good enough for them.
I know it isn't directly answering your question but I'm sure it differs by each airline. SWA and Fedex would probably not know or care about kidney stones/high cholesterol if you have a 1st class; that is a guess though, I'm just a pilot and not a People Department hiring guru.