I'm not debating the importance of states' rights. However, the Civil War was not about states' rights. It was about the South's willingness to destroy the country in order to maintain their ability to own other human beings.
In a TL where Virginia gets rid of slavery, I think that states' rights would evolve in a different direction, since the issue of slavery would not remain at the heart of the issue. The fact that the two national party system (Democrats vs. Whigs) is also maintained will help to keep slavery out of states' rights equation.
With slavery existing only in a minority of states, I don't think that the South will make a united stand to continue to push it west. If slavery is a regional thing that only remains in the Deep South (where its widely seen as necessary to provide cheap cotton to rising American industrial might) then it would eventually end, but with legislative action, not in a fit of attempted national suicide.
States' rights will remain a compelling issue, but without being explicitly linked to the slavery issue the issue will not be settled by civil war.
Agreed, it was slavery that made the issue of "state's rights" to become violent. In fact, without slavery "state's rights" might never become an issue.