As someone who occasionally performs weddings, I can say that the law (in my state) requires that the wedding be witnessed by two people and that there must be some sort of ceremony that they can describe if asked to at a later date.
I insist that a wedding I perform must have at least three components: statement of purpose, consent, and pronouncement.
That is, I will start by saying what we're all doing there, if for no other reason then to make sure that everyone knows we've started, and so that the witnesses can't claim that they didn't know it was a wedding. I will ask both parties if they agree to be married. This may seem silly in this day and age, but there was a time when people were fairly regularly forced to get married against their will, and I think we need to still check in. And then I need to pronounce them married.
I usually strongly urge people to take some sort of vow, but it's not strictly necessary.
Everything else, processionals, prayers, readings, exchanges of rings, kissing, breaking glass, lighting candles, tying hands, jumping brooms or anything else you can invent is all purely optional.
ETA: what I came in to say, and forgot about, is that by my guidelines above, the only thing that seemed missing to me as consent of the two grooms, though perhaps that's implicit in the ring exchange.