The exact quote:
"if we can redefine marriage as between two men or two women or any other way based on social pressures as opposed to between a man and a woman, we will continue to redefine it in any way that we wish, which is a slippery slope with a disastrous ending, as witnessed in the dramatic fall of the Roman Empire."
Well, there were some changes to marriage in Late Antiquity. For instance, a law under Valentinian forbade marriage between provincials and barbarians. There was also more Christian influence on marriage laws, such as Theodosius forbidding Christian-Jewish marriages.
However, based upon your quote this isn't what Carson is thinking about. He is probably trying to blame the fall of Rome on moral or religious grounds - which, incidentally, is a very old phenomenon. This moralising and blaming was happening even as the Empire declined. Augustine's City of God was, in part, a defense against the idea that Christianity was to blame for Rome's decline.