In theory, yes, but in practice the long-established precedent is that the current President's administration does not prosecute the crimes of the previous one.
That prevents each Administration from being a witch-hunt of the previous one, and eliminates a cycle of vengeance that would be counter-productive (or so those on the inside argue).
That's what the Hillary Clinton email scandal is all about. The Republicans are desperately trying to tag her with the onus of having concealed and destroyed emails because for at least six years the entire executive branch illegally conducted its communications outside of the US federal email system, and then destroyed all of the 60 million emails they ran through the RNC and GWB servers.
The next President can drop the hammer on every member of the Bush Administration for deliberately violating the Presidential Records Act, because it has a long statute of limitations. Everyone tainted by it will never again serve in government.
So the GOP is desperately trying to smear Mrs. Clinton, not just for immediate gain but to dissuade her from hitting the administratively competent wing of the Republican Party with the ban-hammer, ensuring that they will never again govern effectively. If she does it, you can bet they'll do it to her, should they ever get that chance again. But it's not looking like they will ever get that chance again, and prosecuting the crimes of the Bush Administration is one way to make sure of it.
Edit: here is a tangentally related article that notes how not one President has prosecuted another one:
http://www.realclearpolitics.com/articles/2009/05/03/look_to_the_future_not_the_past_96313.html
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