In order to build up to true civil war and not just rioting, something would have to unite the causes of everyone who’s mad at the government. Civil rights marchers would have to give up on non-violent protest and follow someone more firebrand like Malcolm X en masse. Young people would have to be a lot more worked up about Vietnam - say, if lots more people were sent there and either were butchered or forced to butcher and the young knew full well it was happening. Women would have to be even more angry than the bra-burning stereotypes and become militant.
One possibility: Nixon beats JFK but by the narrowest of margins, and it turns into something like Bush v Gore and leads to violent protests. The aftermath of the decision, one that puts Nixon in the White House, makes him the paranoid power-seeker that we know from OTL. He handles Cuba masterfully in the 1960s and is re-elected resoundingly, escalating Vietnamcat the same time (thus dampening Oswald’s motivation to shoot him.)
He takes a hard line on civil rights (the 1960 riots change his mind and lead him to oppose the movement.) This galvanizes the movement into a paramilitary organization. People back from Vietnam join. People who don’t want to go to Vietnam join. “Peace and love” becomes a bunch of pissed-off people with guns.
Meanwhile, Nixon is untouchable. It’s even been suggested that he should run for a third term despite the Constitution limiting him to two. Nixon hasn’t suggested at this point that he’s down for running in 1968 and daring the Supreme Court to stop him, but polls indicate he would win a third term handily.
Meanwhile, a Supreme Court case on marriage equality is delayed, and before it can be undertaken, Mildred Loving is found dead.
Sounds like a dystopian TL.