A careful reading of the U.S. Constitution reveals that the majority of the clauses that establish rights actually forbid the U.S. federal government from performing certain actions, period, rather than forbid it from performing those actions or not, conditional on citizenship.
The 4th Amendment protections against unreasonable searches and seizures apply to everyone, everywhere. The U.S. government must always get a warrant, regardless of borders.
U.S. v. Verdugo-Urquidez (1990) [0] established precedent stating otherwise, but I agree with Brennan's dissent and Stevens' concurrence. The majority opinion basically said it would be too difficult for the U.S. to project its power overseas if it still had to obey its own laws there. I don't find political expedience to be an acceptable argument for voiding anyone's rights.
Of course, outside the territorial jurisdiction of the U.S., those protections are unenforceable in practice regardless, because the target likely cannot access any court that might be able to grant relief. It might be interesting for a foreign national to ask to enter the U.S. for the purpose of petitioning for redress of grievances against the U.S. in a U.S. court.
A foreign state need not petition for statehood. It would be sufficient to establish by treaty a court wherein a foreign person may sue a U.S. government and its agents for violations of U.S. law. That would be a double-edged sword, of course. It might also allow a U.S. entity to sue foreign persons for violations of U.S. laws in that foreign state's jurisdiction. (I'm thinking specifically about intellectual property suits.)
[0] https://en.wikipedia.org/wiki/United_States_v._Verdugo-Urqui...