I would allow rescheduling, but be sure to use a different exam.
As some other answers indicated, one probably does not want to get embroiled in the details of the student's medical situation, and it may be inappropriate, besides. (I think it would be illegal in the U.S. to ask for details beyond a general note from the doctor.) Thus, one simply cannot have the information to know whether missing the exam was frivolous or not.
With regard to communication, yes, in principle one imagines that the student would have been able to contact you in advance to explain that (s)he'd miss the exam. But, again, without prying, one cannot know.
Yes, the odds may be good that the situation is due to student negligence than medical emergencies... and we should hope so, in the larger scheme of things. But, since we cannot be sure, my choice would be to treat the situation as a medical emergency, whose details I will not know. And allow further exam retake.
As to whether allowing further exam retakes is fair to the students who've managed to do things on schedule... if there was a genuine medical emergency, we are attempting to compare incomparables. If not, my consolation is that, in my experience, students who miss exams due to their own negligence or disorganization similarly fail to "profit" by extra chances.
Thus, I take claims of medical/personal emergencies at face value, and do not restrict re-takes of exams.