What can Mueller do if he finds evidence of criminality involving the president? He can and will (as authorized by Department of Justice regulations) file a report on his findings with the attorney general (or, since Attorney General Sessions is, in this case, recused, with the deputy attorney general, Rod Rosenstein). Rosenstein will then be faced with the important decision of whether and how to make that report public—whether to convey it to Congress or not; whether to release it publicly or not. The regulations are so vague (they say only that he “may determine that public release of these reports would be in the public interest, to the extent that release would comply with applicable legal restrictions”) that they, in effect, give Rosenstein plenary discretion to do whatever he thinks is in the best interest of the country.
So, every time you read about the threat to fire Mueller, remember this—the critical actor in most future scenarios is not Mueller, but Rosenstein. Knowing Rosenstein personally, I have high confidence that he will make what he thinks is the best decision for the country—the same may not be true of his replacement (or of the replacement attorney general, should Sessions be fired). That, of course, is why the highly dubious “secret memo” prepared by House Republicans reportedly targets Rosenstein—even though he is a Trump appointee who advocated firing Comey, Trump supporters fear he will follow the rule of law.