Is it appropriate to ask one of my thesis-committee members for a postdoc position?
I don't see any reason not to ask. I think that any "conflict" because he will review your dissertation would be minor. A post-doc position is a two-way affair in which you both give and get, so it isn't like pandering on the one hand or asking a favor on the other. 

If your general relationship with the person is good, and you want to work in that area, then yes, ask. But be sure, before you ask, that you wouldn't be rejected out of hand. You need at least to know that you would get serious consideration even if you aren't accepted. An informal explorative meeting might be one way to broach the subject, followed up with a more formal (CV, SOP, ...) application. 

On the other hand, there is also something to be gained from working at a different institution, with new colleagues and new ideas. Don't neglect to look for alternatives in other places. 

But also note that this answer may not apply universally. My experience is in the US in mathematics and CS.