It's interesting to wonder what would have happened had the two split amicably, but I tend to think that the two nations would have re-joined after some time had passed or would have become very close diplomatically.
Clearly, slavery was a dying instutition and once it died out in the South (only a matter of a few decades longer, I suspect), there was little conflict between the two. There is just too much in common between the South and the North for centuries-long hostility between the two.
As far as the 20th Century goes, don't forget that the United States and Great Britain were obviously two very bitter enemies, even as late as the Civil War (Britain was very close to entering the war on the side of the Confederacy), but put all that aside for the wars in Europe, and are now best buddies. I tend to think that a similar course would have been followed between the CSA and the USA, though obviously this is all speculation.
Had they split amicably in 1861, your scenario might have worked out.
Had the South won, say in 1863, it is far more likely that major bitterness and unending conflict would have resulted.
The problem, of course, is that as Lincoln put it, geography ensures that the states could not physically separate. Suddenly all conflicts that previously were dealt with between States would be elevated to international disagreements. Military resolution of arguments becomes possible and attractive to extremists on both sides.