Actually you are the one who is mistaken. There is no consitutional right to privacy. My views on the act in question aren't relevant to a constitutional discussion. (but if you must know I think it's vile, immoral and disgusting.)
But there is. You may not like that the Court is Constitutionally given the necessary power to interpret the Constitution -- but they have been -- and therefore their decisions are the law of the land, unless challenged by the legistlative, executive, and citizens of the US -- primarily by Constitutional amendment.
Procedure was properly followed -- at least to the point that no amendment seems to be arising from the other branches.
If you accept the Constitution, you have to accept the procedures it lays out for its interpretation and enforcement.
Your personal tendency to say "ewwwww" really shouldn't dictate what the law is for consenting adults.
Is it explicitly stated as a power of government?
Nope.
Guess what? That means they don't have any right to infringe on our privacy.