Of course I know it's otherwise protected by the new Respect for Marriage Act and Obergefell, but I happened to be reading about the rather demure P&I Clause in Article 4. and it seemed relevant. IIUC, it's been used to guarantee that "fundamental" rights for a citizen of state A are recognized when travelling to all the other states. Is marriage not considered a fundamental right?
The relevant portion of the U.S. Constitution is the full faith and credit clause (which the Respect for Marriage act is primarily an implementation and enforcement of). This requires states to honor marriages and divorces entered into in different states (subject to limited public policy exceptions that the Respect for Marriage act further limited), and also, for example, paternity determinations and court judgments from other states.
This was one argument among many used to argue that same sex marriages from other states should be recognized in other states. But the U.S. Supreme Court in Obergefell v. Hodges, 576 U.S. 644 (2015), in which it established a right to same sex marriage in all U.S. states, relied instead mostly upon a substantive due process clause analysis similar to that of Loving v. Virginia, 388 U.S. 1 (1967), which held that bans on interracial marriage were unconstitutional.
The Article IV privileges and immunities clause protects very little (mostly the right to interstate travel and the right for an out of state resident to be licensed in a profession in a state).
The Slaughterhouse cases, 83 U.S. 36 (1872), in the late 19th century gave a very crabbed interpretation to the 14th Amendment privileges and immunities clause, limited that only to rights arising from federal citizenship.
The logic of the Slaughterhouse line of cases spilled over into the privileges and immunities clause of Article IV of the U.S. Constitution. The Article IV privileges and immunities clause applies to rights specific to state citizenship to prevent those rights from being denied to out of state U.S. citizens. But those rights are few and far between, and court crafted exceptions have been carved out in cases where it does exist (hunting and fishing license fees and college tuition, for example).