"The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people."
Going to go out on a limb here and say the very fact the FCC exists leads one to assume this is one of things delegated to the federal government -- at least in their opinion.
Pai has specifically claimed the FCC can pre-empt State-level Net Neutrality initiatives.
AFAIK, however, that's up to an appellate court to decide, which hasn't happened yet. (Another comment in another thread cites the recent case the FCC lost regarding its ability to pre-empt municipal broadband — see TN, NC v. FCC — which is probably somewhat dispositive.)
EDIT: also: "delegated to the United States by the Constitution". Note the emphasis. The FCC can assert all the authority it wants, but absent an Amendment to that effect, or a specific ruling on that question by a "court of competent authority", I don't believe the matter is nearly as settled as you suggest.
IANAL, but near the end he says he'll be including it as a condition to certain things that the state has control over. maybe that's how they get around that issue?