Parts of it clearly do. States are prohibited from maintaining a Navy or Army, they are prohibited from collecting tarrifs on goods exported to or imported from other states. They are forbidden from making treaties with foreign nations.
But if you mean the Bill of Rights, that is subject to debate. Except for the first amendment, which clearly states *Congress* shall make no law.
However the original intent of the 14th amendment was to apply the BoR to the states. That's how it was understood by both proponents and opponents. Only some "Men in Black" who didn't like that notion, read it differently.
The text of 14A most certainly does not "incorporate" the BOR as applying against the states. Any such incorporation ("selective incorporation") was achieved via judicial activism and findings of "substantive due process" - yet another fiction.
My understanding of the original intent of 14A was to force states to treat citizens equally (blacks and whites for example) - even in a state's appication and enforcement of seemingly unfair state law (no person shall hold office unless he believes in God for example).