There would definitely have been a second war.
There were several "sectional crises" that eventually led up to the war, all of which had to do with western expansion and the status of slaves in the new states.
Had the South won, westward expansion would still have been an issue, and now there would have been two nations involved, with pre-existing bad blood between them. The war would definitely have begun anew, in the west -- probably within 5 years of the end of the previous one.
This war would NOT have lasted into the 20th Century, because the preponderance of Northern industrial might (which effectively won the first time) would still be relevant.
The Union badly defeated the Confederacy in the western war almost from the very beginning. I think the primary reason is that the western war spanned a huge territory, and depended much more on logistics. The Union had much better logistical capacity ... and they also had Gen. Thomas to figure out how to use it. Logistics would have defeated the South in a subsequent western war as well.
I just re-watched this series a few weeks ago. One statement really struck me this time around. Shelby Foote stated that the Union won the war "without really trying." He stated the Union had vast resources that it didn't even use during war and that the lives of the Northerners weren't really affected at all by the war effort. The only sacrifice that the citizens of the North made was their young men (which, of course, isn't to be minimized).