I'm not sure about the public backlash, but I'm 100% certain that Mueller has a plan to release whatever findings he has uncovered so far to the relevant state level attorney generals (attorneys general?). Most notably, New York, where AG Schneiderman has already indicated his distaste for all things Donald Trump. Of course, presidential pardons do not apply to convictions levied in state courts, so he may plan to route some prosecutions that way already. But yeah, especially once Mueller indicts anyone in Trump's family (we're looking at you first, kid Kushner), I expect him to be fired within a couple of days, tops.
Attorneys general, yes. It's interesting to me because Trump has long functioned by bullying people and burying them in bullshit, but he's traditionally picked fights with people he knows he can beat. Now that he's literally the biggest target in the world he's no longer getting to pick. My understanding is that Mueller is wholly beholden to the Attorney General, or in this case, the deputy Attorney General. What that means is anybody's guess at this point.