Indeed it does, and it's protected from governmental infringement by both the 9th and 10th amendments. When the Constitution and Bill of Rights were written, marriage was not a concern of government at least not of legislautres. It was a concern mainly of the churches, and secondarily of the courts. Hence "common law marriage", which is marriage outside both church and state.
The same is true of birth records. Those were strictly private and church business until much more recently than the late 1700s.
Yes, the 4th Amendment is important here. I've been quoting it to other libertarians today. The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures
Now from that you deduce that we have a Law Free Private Zone, or the Mythical Right To Privacy?
Good golly, people, first I see all these Libertarians running around like chickens with their heads cut off shreaking about this being the beginning of an Orwellian conspriracy, but the 4th Amendment should calm the fears of that conspiracy. So you read that, quote from it, yet still can't see the truth!
The operative word in the Amendment is unreasonable meaning get a bench warrent. Are you worried about a judge swearing out a warrent for a Class C misdemeanor of Sodomy in Texas? No. Now if your beloved Right To Privacy isn't fought against by conservatives, all the issues Santorum spoke of, in addition to dealing drugs, prostitution, you name it-become sealed off from Justice. Where do you draw the line? You CAN'T when you add to the Constitution this canard of an arguement.