Second, since the investigation would still be open, we might see the investigation continue but without a special counsel. That’s partially what has already happened with respect to the investigation of Trump’s attorney, Michael Cohen. That investigation is now being handled by the U.S. Attorney’s Office in the Southern District of New York—a regular unit of the Department of Justice. If Mueller were fired, the ongoing pieces of his investigation (into Manafort, Russian social-media interference, etc.) could also get farmed out to various U.S. attorneys’, or a Department of Justice, litigating division to pursue. If that happened, the investigations might lose some coordination, but they would continue apace.
Finally, and most problematically, whoever fired Mueller could also order that the investigation be closed out. On the merits, this would be a dubious act, in the sense that there is no actual basis in fact for shutting the investigation down. In “real life,” investigations are closed only when their original predication is disproved or when they’ve reached an investigative dead end and can go no further; or, sometimes, when the resources necessary to continue exceed the apparent significance of the crimes under investigation. Whoever obeyed Trump’s order to fire Mueller might also pull the political trigger and shut down the investigation and brave the political storm that would follow.
But even that would not end the matter—at least not necessarily.
The U.S. government may properly share information collected during an investigation when authorized by statute or by a court. It is quite possible (indeed, if public reports are to be credited, highly likely) that the special counsel has already shared relevant investigative information regarding possible criminal violations of New York State law with the attorney general of New York. No action by the federal government could limit that state official, nor could any action prevent him from continuing the investigation. To be sure, not all of the matters under investigation by the special counsel are also potential violations of state law—but many of them appear to be.
And what about Rosenstein? What if he is fired, but Mueller is not? The idea here, of course, is that someone who replaced Rosenstein and is more loyal to Trump might limit the special counsel’s investigation—might, for example, tell him that he could no longer look at the financial records of the Trump Organization or might order him to stop all inquiries into the Trump Tower meeting between the Trump campaign and purported agents of the Russian government. This would be an effort to control the Mueller investigation indirectly, without actually firing him—and it might be that some would judge firing Rosenstein less politically tumultuous than firing Mueller himself.