The most important statute here seems to be 28 USC §508(a), which directly adresses the position of Attorney General:
In case of a vacancy in the office of Attorney General, or of his absence or disability, the Deputy Attorney General may exercise all the duties of that office, and for the purpose of section 3345 of title 5 the Deputy Attorney General is the first assistant to the Attorney General.
Note that it does not matter under this statue what caused the vacancy. It doesn’t matter whether Sessions was fired or resigned. But it also references the Federal Vacancy Reform Act, 5 USC §3345, which applies generally to officers of Executive agencies. But per §3347 of that law:
(a) Sections 3345 and 3346 are the exclusive means for temporarily authorizing an acting official to perform the functions and duties of any office of an Executive Agency (including the Executive Office of the President, and other than the Government Accountability Office) for which appointment is required to be made by the President, by and with the advice and consent of the Senate, unless—
(1) a statutory provision expressly—
(A) authorizes the President, a court, or the head of an Executive department, to designate an officer or employee to perform the functions and duties of a specified office temporarily in an acting capacity; or
(B) designates an officer or employee to perform the functions and duties of a specified office temporarily in an acting capacity;
So per 5 USC §3347(a)(1)(B), since 28 USC §508(a) is a statutory provision designating who may temporarily act as AG, it should still apply.
Normally, since 28 USC §508 is both more restrictive, and specifically applies to the office of Attorney General, I think it would be considered controlling here. But since it directly references 5 USC §3345, it seems to anticipate that that statute still applies, as well.
So arguably (but I think debateably), the other options available under 3345 for temporary appointments may also be available to the president. The only one here which could allow Whitaker, is 5 USC 3345(a)(3), which allows temporary appointment of any agency employee, who has served for at least 90 days, at pay grade GS-15 or higher.
The first objection to that might be that 3345(a) only applies when an officer “dies, resigns, or is otherwise unable to perform the functions and duties of the office”. If congress had meant this to apply in the event of any vacancy, they could have used much simpler language (like, “in the event of any vacany….”). A seperate clause, 3345(c)(2), specifies that “the expiration of a term of office” is considered an “inability to perform” under this section. But nothing clearly says that this can be applied in the case of a firing. This question appears to be unresovled as of yet by the courts. The issue was raised recently in Hamel vs. Veterans Affairs, but that case was dismissed without any rulings being made after the appointee being challenged was confirmed by the Senate.
Second, it can be argued that even if congress intended to allow such an appointment, it may not itself legislatively undermine it’s own advice and consent power in this way. This position was taken by Justice Clarence Thomas in a recent opinion (pdf), in which he argued:
That the Senate voluntarily relinquished its advice-and-consent power in the FVRA does not make this end-run around the Appointments Clause constitutional. The Clause, like all of the Constitution’s structural provisions, “is designed first and foremost not to look after the interests of the respective branches, but to protect individual liberty” NLRB v.Noel Canning, 573 U. S (2014)….The Judicial Branch must be most vigilant in guarding the separation between the political powers precisely when those powers collude to avoid the structural constraints of our Constitution
It should be noted though, that this was the opinion in this case of only Thomas. The majority in this case ruled that the appointment in question violated the FVRA, and thus they had no need to adress any constitutional issue, so we don’t know their opinions on this. But two dissenting justices (Sotomayor and Ginsberg), held that the appointment was legal, and thus they must have seen no constitutional issue with an appointee under 3345(a)(3), who had not previously been confirmed by the Senate, who served for almost 3 years as NLRB general counsel.
Ultimately, as none of these issues has yet been clearly resoved by the courts, the Whitaker appointment will likely need to be litigated. I favor holding 28 USC §508 to be controlling in the case of the Attorney General, and also believe a firing is not a resignation, or otherwise an “inability to perform”. That would mean the Deputy Attorney General Rod Rosenstein should be the legal acting Attorney General, and the next in line would be Associate Attorney General Jesse Panuccio. Some more in depth treatement of these legal issues can be found here.
A final point though, regardless of how these questions are resolved, if the President’s purpose in this forced resignation and appointment is to interfere with and obstruct the Mueller investigation, then that in itself makes the appointment illegal, as obstruction of justice. And there are plenty of details as well about Whitaker, which I won’t go into now, which suggest that this is the case, and that he would be fundamentally unqualified for the position, anyway.