Is a professor allowed to require me to give him permission to speak with my doctor before letting me make up an exam?
That sounds like a big no to me, but that comes from Germany with very strict rules on privacy. Your medical condition is your thing, and yours alone. If you have a document that certifies your illnesss, that should be enough.

Check your university policies, what you need to provide in case off illness HAS to be written down somewhere. If it states there that a written thing is enough, it is enough. Point your professor to that document, if that does not help talk to faculty cousellors. They should be able to help you.