For reference, here's the amendment, with the pertinent clause highlighted:Plenty of Constitutional scholars have pointed out that the amendment, proposed in 1866 and ratified two years later, was designed specifically to grant full citizenship rights to newly-freed black slaves; now it's being used to say that denying SS couples the 'right' to marry is unconstitutional, because it denies them the equal protection of the law. But that's not really true; if anything, it denies them the benefits provided by the government to married couples – it can't deny them equal protection of the law, because there is no constitutionally-protected right to marriage of any kind.
1. All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.
If SCOTUS somehow finds that the Equal Protection clause grants a constitutionally-protected right to SSM, it must also provide the same right to any other form of marriage – traditional, polygamy, polyandry, child, and so forth; if it does not, then it is in violation of the Equal Protection clause of the 14th Amendment.
And if SCOTUS finds (or creates) a 'right to marriage', that's going to open up another can of worms.
Hypothetical: Person A asks Person B to marry them. Person B refuses. Person A sues Person B in federal court for violating their constitutionally-protected right to marriage.
Hypothetical: Persons A and B are married. Person A files for divorce. Person B doesn't want to divorce. Person B sues Person A in federal court for attempting to deny them their constitutionally-protected right to marriage.
People may scoff at those examples, saying that these laws would only apply to people who love each other; in fact, that has been part of the argument over SSM – that two people who love each other ought to have the right to marry. But this would require SCOTUS or the lower courts to cobble together a legal definition of 'love' – something even non-legal folks have never been able to do adequately. And I doubt I need remind any of my readers that the majority of divorced people are those who professed to love their partner when they got married.
And if you're tempted to scoff at my hypotheticals up above as something that would never happen, remember that less than two generations ago SSM was also something that would never happen.