I don't think most Americans would think of the situation in terms of "natural rights." They might think of Christian morality, in terms of how we should treat other people, but I don't think the Bible has much to say about search and seizure, or electronic privacy, etc.
And for myself, I don't like to argue in terms of any sort of thing, "natural rights" or "God's law" or whatever, that a doctor couldn't find if he cut me open and poked around. To me, the ability to act is the only thing that exists, and "rights" are just people in a community agreeing to use collective force to enforce whatever they decide to call "rights." The idea that the community as a whole can be bound to limit its ability to act with respect to people outside that compact is to me not a sensible thing.
And I think that while most Americans wouldn't phrase it precisely in that way, their thinking is along similar lines. We're a country that strenuously reserves the right to do whatever the hell we want, bound only by our own collective conscience. We want to do right by people, but we get to be the final arbiters of what's right and what is justified.