I can't help you on specific examples of ideological divides but there's been a number of mutually agreed national partitions. Norway/Sweden (non-violent), Singapore/Malaysia (racial riots, no civil war. Singapore is now one of the richest countries in the world and kicks America's ass in a variety of metrics), Czechoslovakia (non-violent), Cyprus (part of cease-fire) are a few examples.
Those are all the divisions on national and ethnic lines (except Cyprus; seriously dude? Might as well put Abkhazia and South Ossetia on that list). There's not a single example of political divisions, and with good reason, as the only ones I can think of off the top of my head are Vietnam and Korea, neither of which are something you'd want to emulate. Although I suppose North Ireland maybe counts? Not that you'd want to emulate
that. Moreover, as someone whose family is from the former Yugoslavia I have a rather strong aversion to the breakup of states along lines anyway, as they tend to leave the individual states less than the sum of their parts... But regardless, do you really think that demands that a breakup will reduce division and bloodshed? That will come to nonsense as soon as the all-too-important question "who gets what?" is asked. Divide exclusively on state lines? If so, on which ones, and on what metric? And if not, how to handle the fact that geography is complicated? Does North New York get to secede and join Republican land? How about North California? Can Austin join the Democrats? What kind of broken monstrosity will result? And once you remember that not all land is made equally, and some areas are rich in oil and resources... Must we relearn the mistakes of Bleeding Kansas?
To give a less abstract and slightly more recent example, Jaywalking. In the early 1900s you had a bunch of people moving from rural to urban and they didn't understand city etiquette. As a result they'd walk out into the roads. This isn't a huge problem AFAIK jaywalking is legal in most nations on Earth barring super high capacity roads like highways. But in America as a result of the tensions caused by migration they not only criminalized it, they named the law after "jay" which at the time was derogatory slang for those same rural migrants (hence "jaywalking").
This is definitely different from the story I (as an urban dweller) have been told. It was the automobile companies that popularized jaywalking laws, and it has always been in New York, which is steadfastly anti-car in orientation (and has remained so since the 1900s), which has the most weakly enforced jaywalking laws in the nation. In fact, I once read a local magazine discussing the best ways to cross, leaving a space at the end for "The Tourist" method, which involved standing at the crosswalk, waiting for the light to change "as if this was freaking Denmark or something." And on a personal level, it's the people who come from more rural, suburban, or car-focused locations (LA, for example) which are terrified of crossing at the "wrong" time, and people from my area (NYC) who found that positively ridiculous. Attitudes and Response to Jaywalking is certainly
correlated (although not perfectly) with the urban/rural divide, but the
causation is a location's relationship to cars.
On an unrelated but very relevant to thread note,
John McCain is apparently undergoing a surgery to remove a blood clot from his left eye, and this has imperiled the repeal effort.