It's complicated, as usual. First, most GME funding is from Medicare, not Medicaid (although Medicaid does have some GME funding, so complicated).
What the current Admin can and can't do is very fluid. And even if they "can't" do something, they can do it and wait for it to play out in court. Which can take long enough such that it no longer matters.
How much of the rules of all of this are actually in a congressional law vs how much were created by the rule-making process I don't know. But most of it appears in the
Federal Register, so I expect its mostly rules. Which means that they could change at any time without input from Congress, I believe.
Actually, the law is in the SSA section 1886 paragraph H. And there, it says:
(h)(A) Rules.—The Secretary shall establish rules consistent with this paragraph for the computation of the number of full-time-equivalent residents in an approved medical residency training program.
So you see, the HHS secretary can change this any time they wish. POOF!