One of the beautiful pieces of the Constitution is that it's allowed to contradict itself. The more specific statement wins out (for example, a section banning gay marriage would beat out a section that said that all people were guaranteed equal rights)
And the most beautiful part of the Constitution is the checks and balances. For instance, the Executive branch is supposed to enforce laws from the most generic to the most specific, as you've indicated. However the Supreme Court Justices are supposed to use their brains, and realize that a law banning gay marriage, written into the Constitution or not, is inherently unconstitutional because the Equal Protection Clause takes precedent.
