yep, a CSA and a USA, living side by side is almost a foregone conclusion in the event of stalemate.
The question is what happens from there. Do we have enmity between the two states perpetuating on for all eternity, with likelihood of another war sometime down the road? or do we have enmity for a while with trade going on between the two and gradual movement toward acceptance/friendship? I vote for the latter. I believe both sides will remain in a military stance for a while, wary of the other, but the CSA was never about conquest of the North/USA, and the USA was all about there not being a CSA. Once they accept the existence of the CSA, I think dreams of a reconquest will slowly fade. The two make natural trade partners, so there will be economic intercourse between the two. As long as they remain wary of each other, both sides will maintain a military. It may not be as big a part of the GNP as the European states, but it won't be as small as the OTL USA of the late 1800's.
The north will likely rebound economically first. The south doesn't really have the resources to industrialize, at least not rapidly, so it's growth will be stunted. Slavery will inevitably hold the south back, economically, and sooner or later will come up as an issue again. This means the USA will be able to exploit a western expansion, while the CSA remains hemmed in it's 1/4 of the OTL USA. By the time the CSA has recovered and can think about expansion into the carribean, slavery is going to be on the decline (I think that movement is inevitable), so there won't be the incentive to build up a navy/army capable of expanding.
Bottom line, I think the North ends up as a powerful economic Big Brother to an inferior economic South, and depending on how things shake out, the border states may start thinking about reintegrating with the North. What will hold this back is the degree to which the North sticks with the OTL centralized national power at the expense of the states. While slavery was, bottom line the cause of the war, states rights were a very real issue, and OTL, states rights took a back seat. IF the CSA remains a confederation rather than a central gov't dominating, states may be happy being more independent rather than giving up their fiefdoms to reintegrate with the North. OR, the North may not stay with as powerful a central gov't. Having recognized the right of secession, they may have to recognize states rights to keep states from leaving, or to entice border states to come back. That whole dynamic may be the friction that leads to round two, and under my scenario, time is the friend of the North, militarily. The situation is similar to France/Germany of the same time period. France/CSA will grow, but lose ground to a faster growing North/Germany. Turtledove's vision is all wet, IMO, and was pretty unbelievable.