Spoiler alert: same-sex marriage is now legal in the U.S. This happened on June 26, 2015 in a Supreme Court ruling of 5-4. Here's why:
"No union is more profound than marriage, for it embodies the highest ideals of love, fidelity, devotion, sacrifice, and family. In forming a marital union, two people become something greater than they once were. Their hope is not to be condemned to live in loneliness, excluded from one of civilization's oldest institutions. They ask for equal dignity in the eyes of the law. The Constitution grants them that right." (Source)
There's still plenty of debate, though. For example, Justice John Roberts claims that same-sex marriage has nothing to do with the Constitution. Justice Scalia said the decision was a "threat to American democracy" and a "judicial Putsch" (source).
Do they have a case? Let's get into the brains of the five judges who voted in favor, just to be sure.
Justice Kennedy referenced Loving v. Virginia (1967) multiple times. This court case overturned state bans on interracial marriage, ruling that "freedom to marry" was an essential civil right. Most people would agree that hey, if race shouldn't determine whether people should get married, neither should gender. Denying anyone a right isn't equal treatment under the law. Straight up.
Specifically in Loving v. Virginia, the Court held that states could not ban interracial marriage since "the freedom to marry has long been recognized as one of the vital personal rights essential to the orderly pursuit of happiness by free men."
To "deny this fundamental freedom on so unsupportable a basis as racial classifications," Chief Justice Earl Warren wrote, was "directly subversive of the principle of equality at the heart of the Fourteenth Amendment." In short, "under our Constitution, the freedom to marry, or not marry, a person of another race resides with the individual, and cannot be infringed by the State."
Good ol' Justice Kennedy.
In Obergefell v. Hodges, the Due Process Clause is exactly what Judge Kennedy ended up referencing:
Under the Due Process Clause of the Fourteenth Amendment, no State shall 'deprive any person of life, liberty, or property, without due process of law.' The fundamental liberties protected by this Clause include most of the rights enumerated in the Bill of Rights
So basically, denying marriage to anyone is depriving a bunch of American citizens the freedom to marry without due process of law. Obergefell v. Hodges is that due process. And as five out of nine Supreme Court judges decided, denying gay couples the right to marriage is not equal treatment under the law.
So, back to Justice John Roberts' claim that marriage is not a Constitutional issue. Is it? Well, maybe it wasn't when the very first cavemen were getting hitched and buying their very first caves together.
Let's state the obvious. Marriage wasn't a Constitutional issue before the Constitution existed. Nowadays, the government gives out marriage licenses and provides tax benefits and incentives to married couples. The government has definitely involved itself in the issue of marriage.
But is marriage a right or a privilege? The debate rages on, but five out of nine Supreme Court justices definitely say it's a right—a right that can't be denied to someone based on sexual orientation.