Probably worth mention, the US doesn't appear to have signed that and for that matter many of the signatories don't allow gay marriage either, there's about 10 countries, starting with the netherlands in 2001 that allow it, and two who allow it in some places, the US is one and Mexico is the other, there's a roughly equal number where it carries the death penalty. and there are 192 nations in the UN, so we're not in much need to worry about violations

Anyway the issue on 16 is "Men and Women" since everywhere else it says 'Everyone' 'all' or 'No one' and such, but regardless, the clear spirit, not too mention the standard interpretation, are that gay marriage is not banned nor granted by article 16. Of course the UDHR is so much worthless paper anyway. It tends to get pretty specific too, wasn't written on a napkin, article two has a whole laundry list of "such as race, colour, sex, language, religion, political or other opinion, national or social origin, property, birth or other status" regarding who the rights apply to. There's also the generally futility of amending it anyway, since it is a non-binding measure, and there is, in article 29. a rather nagging "meeting the just requirements of morality, public order and the general welfare in a democratic society." which is regularly cited by countries who pay the UHDR lip service. To the best of my knowledge our only violation to date is actually article 16, the marriage clause, since some states kept bans on interracial marriage for a while, and argues that the clause didn't say you had to permit interracial marriage, only that you couldn't deny someone of a given race the right to marry. There's also the article 5 torture clause but that's another issue.

The problem is lots of people go around declaring what something means, so courts decide, and I don't think you have made the case that the US and 181 of 192 other countries are not in violation of article 16.