The noun phrase "doctor's appointment" is possessive. It is an appointment that the doctor possesses, not the patient.
So technically when a patient is speaking about his or her appointment with a doctor, the patient should say I have a doctor appointment. "Doctor" modifies "appointment" and tells what kind of appointment the patient has.
A "doctor's appointment" could be an appointment the doctor has with his girlfriend, his psychiatrist, or his dentist (or any other person for that matter).
Think of it this way, do you ever hear a person say I have a dentist's appointment?
Unfortunately, "doctor's appointment" still seems to be the preferred phrasing as exhibited by the screen shots and by Grammar Girl's take on the situation.
I've looked this up in a bunch of different places. I found conflicting answering, none of which seemed definitively convincing. I prefer "doctor appointment" because it makes more sense to me (it's my appointment with the doctor, so doctor is modifying appointment), but "doctor's appointment" seems to be more common.Sorry I can't be more helpful!
How will people ever get it straight when a site named Health in Plain English can't even get it?