A constitutional amendment defining marriage as between one man and one woman would take ratification of 2/3 of the states. Would 2/3 of the states ratify such an Amendment? If just the states that voted against gay marriage already voted for ratification it would pass. It would pass in states that haven't voted against gay marriage but had it imposed on them.
If it is true that the "vast majority" of people support same sex marriage, then there's no problem whatsoever with putting it up as a Constitutional Amendment.
Correction- An amendment would take approval of 2/3rds of BOTH houses of Congress, and then 3/4 of the states.
More to the point, amending the constitution shouldn't be because you don't like the law. I would recommend that you look up the whole prohibition thingee, and see how well that worked out. A bunch of dumb, bible thumping assholes made alcohol illegal by playing on anti-immigrant sentiment during WWI, got a dumb amendment passed, and 15 years later, everyone had to admit they screwed up and repealed it.
There is no law, nor should there be, prohibiting homosexual conduct. Such a law would be abhorrent and reprehensible. Forcing the majority to recognize the legitimacy of such relationships are the issue, not the existence of such relationships.