Gee....if you look hard enough it is in there somewhere
Lets try the 14th Amendment
All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.
So, right now we have laws which prohibit homosexuals from marrying the person of their choice. We also have laws prohibiting them from serving their country.
We also have laws prohibiting heterosexuals from marrying the person of their choice. What if the person I want to marry is already married? A minor child? Dead? The right to marriage has never been for "whomever you choose". It also has never been reserved solely for heterosexuals, as evidenced by the fact that even a gay man could legally marry a woman, but a straight man couldn't legally marry another man.
Don't Ask, Don't Tell prohibits them from serving openly, even though there are plenty of stories of gay soldiers who have come out to their comrades and said nobody had a problem with it. However, there's no "right" to serve in the military. They can deny you entry for a host of reasons, and can kick you out for any host of reasons, too.