The problem with using "impeach" is not that the judicial position must be an elected one; as has been pointed out, holders of positions to which one is appointed (such as federal judges or cabinet secretaries) can also be impeached.
The reason "impeach" does not work is that being impeached is like being indicted: while it may result in a trial, it does not guarantee either conviction or the subsequent penalty.
As the example of Donald Trump shows, one may be impeached more than once, and yet still not be removed from office. It is therefore necessary to use a phrase, although one has a number of options depending on how emphatic or dramatic one wants to be (e.g., "he should resign his position", "he should be removed from office", "he should be thrown off the bench"; etc.)