The Tenth Amendment (Amendment X) to the United States Constitution, which is part of the Bill of Rights, was ratified on December 15, 1791. It expresses the principle of federalism, which undergirds the entire plan of the original Constitution, by stating that the federal government possesses only those powers delegated to it by the Constitution. All remaining powers are reserved for the states or the people. In drafting this amendment, its framers had two purposes in mind: first, as a necessary rule of construction; and second, as a reaffirmation of the nature of the federal system. Marriage is not mentioned in the Constitution, in any way, shape or form, so the STATES define what constitutes a marriage, and the States overwhelmingly have said marriage is a man and a woman, not a couple of sodomites that want to have sex 24/7, and absolutely can not bring children into the world as a result of that sex union. End of story.
No, the states cannot define marriage insofar as said definition becomes an attack on the free exercise of religion. In fact, the free exercise clause as supreme law of the land could (and perhaps would) preclude that being a state’s right.