One factor would be how the separation was accomplished. A peaceful resolution of a situation that usually led to bloody war would be something to be proud of. On the other hand, a defeat sufficiently decisive to let the southern states have their secession could make people question the viability of our republic or succumb to bitterness, frustration, revanchism, or other negative feelings.
Very true. The devil is always in the detail. Even various forms of defeat for the north could have different impacts, including as BreuandGrayl says above possibly resulting in some border states joining a victorious south, or possibly in Kentucky's case being the POD that leads to a southern victory.
The worse case scenario you mentioned, with bitterness, possibly infighting over scapegoats, high level militarism and possibly increased xenophobia is a possibility but hopefully some level of democracy and political stability would survive else the rest of the century, at least, could be nasty for both the rump US and its neighbours.