Read through the Constitution.
You will not find a right to same-sex marriage anywhere in the Constitution, or the Bill of Rights, or the other amendments. Of course, you also won’t find a right to heterosexual marriage.
Check me on this: none of the words or phrases “marriage,” “married,” “family,” “engaged,” “right to marry,” “right to marriage,” “open marriage,” or “it’s complicated” appear anywhere in the document. The generic word “posterity” shows up in the Preamble, but that’s as close as you’ll get to “family”. In other words, establishing a national, uniform marriage regime does not relate to the purposes of the Constitution as initially conceived. Unfortunately, this could have a divisive effect rather than the unifying influence George Washington saw as one of the Constitution's prime benefits. Why? Because except for the legitimate Constitutional question regarding the “full faith and credit” clause in Article IV as to whether states would be required to recognize same-sex marriages from other states, the marriage question need not have been settled at the national level.
Why? Marriage and family are really important, right? Sure. But the Constitution initially had nothing to do with those really important things. Its purpose was to establish a national political order out of a diverse body of sovereign states, not to manage intimate living arrangements for the inhabitants of those states. Towns, cities, and states were already managing intimate affairs like marriage without permission from judges in Washington D.C., or Philadelphia. The roots of the American tradition of self-rule run much deeper than the Constitution.
Alexander Hamilton, advocating ratification in The Federalist Papers, said its “principal purposes” were the provision of common defense from internal and external threats and the management of relations with foreign countries. The Constitution empowered a federal government to protect general or national interests, preserving the liberty of self-rule at the local and regional levels. The brief document laid out what the federal government could and could not do, enumerating the limited powers of each branch. All other powers were left to the states, which each retained authority over their internal affairs to the extent consistent with other constitutional provisions.
The thirteenth and fourteenth Amendments abolishing slavery, providing for equal protection of the laws and guaranteeing citizenship to everyone born in the country regardless of race, increased federal government influence over state affairs after the Civil War. This development and progressive turn-of-the-century politics presaged the further expansion of federal influence accompanying the tumult of the twentieth century: the Great Depression and two world wars necessitating what President Dwight D. Eisenhower described as the “military-industrial complex.” Lyndon B. Johnson’s Civil Rights and “Great Society” legislation also expanded the scope and reach of the federal government and its bureaucratic apparatus, further nationalizing American society.
In addition to abolishing the great evils of slavery and Jim Crow, the nationalization of American society has meant that the most intimate affairs, initially left to states and localities, now dominate politics and legal proceedings at the federal level. Supreme Court decisions like Roe v. Wade in 1973, and recently the Obergefell decision, treat the Constitution as if it were primarily designed to protect citizens from their states and communities, rather than states from encroachment by the federal government. The Supreme Court has appealed to the ethos of the Civil Rights movement, treating the Constitution as a penumbral well from which to draw ever-expanding individual rights, and to revoke states’ authority to democratically define and uphold societal norms related to sexual behavior or the family. It's a move that could have serious social consequences.
Even with the thirteenth and fourteenth Amendments, which began to extend federal government influence in states' domestic affairs to guarantee the rights, privileges, and duties of citizenship to all persons subject to U.S. jurisdiction, the actual Constitution is a different document than the “Constitution” existing in the public imagination. Why? It's largely due to the freewheeling interpretive lens the Supreme Court has applied of late.
Read through the Constitution—the actual one—and see what you think.