Even though the word privacy is not used in the U.S. constitution or its amendments, all of the following amendments are associated with privacy, according to the Supreme Court EXCEPT:
As near as I can tell, it is the 7th although I can't rule out the 14th. The 7th guarantees a trial by jury over a civil dispute exceeding 20 US dollars. It also talks about how the trial is won.
The third is definitely a privacy issue. No soldier can be billeted in a citizen's house without the owner's consent.
The Fourth is even more to the point. A search and seizure without cause violates the fourth.
The 14th is a real devil. It is very long and a lot of behaviour can be hidden under its skirts.
I think the 4th clause of the 14th touches on privacy. It talks about payment of debt arising from putting down an insurrection. Such debt payment shall not be questioned. That means to me that the person applying for relief has a right not to disclose what he did to quell the rebellion and that's a privacy issue.
The 14th is always a handful of law. You may be wrong but the seventh is your only other choice and the burden of proof is on the person bringing the case to trial.
I'd read over clause 4 of the 14th and see what you think. My understanding of the American Constitution is random at best.