Would most PhD supervisors stop working with a student who was unproductive due to clinical depression? Suppose a PhD student got clinically depressed and stopped producing research for a year or two.  

What would most PhD supervisors do in this situation? Would it be considered usual or acceptable to stop working with him?
Good PhD Advisors(tm) would first of all talk to the student about the issue. Large-scale depressions need to be properly treated and respected. The important thing is to find out if it is the academic work that causes the depressions and to take counter-measures if it is.

Now if the depression lasts, is properly treated and puts obtaining the PhD at >90% risk of failure and all possible measures have been taken to help the student (both on the professional level and as far as applicable and feasible, on the personal level), then the student is wasting time in academia. And nothing is more depressing than that. So a Good PhD Advisor(tm) would help the student with establishing a new working life that makes use of the degrees already obtained as much as possible. This is basically the best the advisor can do to pave the way to a happier life for the student. I wouldn't call that "firing" someone.