They can try. But the result will be exactly the same as for every other group of men who tried to do so throughout history.
Reality and right can be very harsh when you put yourself on the wrong side of it.
No, it is the very nature of the male-female relationship that makes it uniquely suited for marriage. Our state laws regarding marriage should reflect the high value that society places on this bedrock institution, and elevating homosexual unions to the same status as marriages runs contrary to the Natural Law.
The only way to prevent these six states that currently allow homosexual marriage from solemnizing any more of them (and any more that ban it in the meantime) is to pass a federal constitutional amendment banning homosexual marriage nationwide. The odds of it succeeding are not very good, but it is the only constitutional way for the federal government to get involved in the marriage business. Time is really of the essence here, however, because once 13 states have legalized homosexual marriage, the battle is lost. You need three-quarters of the states to ratify it.
No, Congress can simply legislate against it. They have all the legitimate authority to do so that they need, since the destruction of the most fundamental unit of self-government violates every single expressed purpose of our Constitution, most notably its crowning purpose: “to secure the Blessings of Liberty to Posterity.”
But if you want to pass an amendment for the purposes of clarification, fine. Do it.
There's still the issue of the existing marriages. Realistically, they would have to be allowed to stand (even Nazi Germany's Nuremburg Laws did not go so far as to dissolve pre-existing Jew-Aryan marriages).