Say I went to a taping of Hell's Kitchen and now I want to spoil the ending because I think Gordon Ramsay discriminates on the basis of race (political speech), can I just violate my NDA? Fox sues me for breach of contract, I lose, and court orders the agreed upon remedy (say I agreed to be fined should I violate terms of the NDA). Is the enforcement of this NDA invalid on first-amendment grounds?

Likely no. Because the restriction of my speech was enforced under the principles of contract law, not under any law abridging freedom of speech.

Applied to this case, the NCAA would be petitioning a court to enforce a provision of it's own law, citing the courts authority to protect the interests of parties who contract. The court is in no way shape or form enforcing provisions of a government statute abridging the freedom of speech. There is no First Amendment test.

Did you mean in tort? Criticism of well known organizations is likely covered speech, but Manziel isn't saying anything about the NCAA. I'm struggling to think of any standing the NCAA would have based on this that would sufficiently show actual injury. The only type of analysis i'd do is through contract law (although it's not apparent Manziel is bound by any such contract).

TL:DR; Enforce "it" under what statute?