The Civil War, was in fact, over states' rights, rather than slavery. It was essentially the only reason, with only a small sector on both sides (abolitionists and people who heavily depended on slaves) who viewed it as such.
The first part is the wish to be able to see where the rights of the slaves was drawn from that of the federal, and to test the limits. This was found in the 10th Amendment of the U.S. Constitution, in quote, "powers not delegated to the United States, nor prohibited by it to the States, are reserved to the States". In this case, the right for a state to cede, as well as their right over whether or not they have slavery or not, simply is their choice, rather than being bound with the federal government and the United States.
However, the U.S. Federal Government and the North did not believe so, and even raised an army to force the south to rejoin. This was their first objective. In fact, slavery was not essentially a main objective until the Emancipation Proclamation was published, which effectively ended slavery only in the South. Any states that were still part of the union that allowed slaves technically still had state's rights over them. Simply going over war over slavery as a subject would have been frowned upon in that day and age, and would not have garnered large amounts of support in any way, shape, or form.
Notice how while slavery is the selling point of the war, it is not exactly the main reason. States individually believe whether or not they are able to first benefit from slavery, and secondly to set whether or not it's own citizens can have slaves.
The next part is to suggest that there was actually not going to be a war if the states did not cede. There would not have been a good reason, as slavery has already been prevalent for years, and to suddenly go to war about it would have put the world, who was watching this interaction, firmly on the side of the South, which the North was looking to avoid.
Slavery was a main part of the Civil War following the Emancipation Proclamation, where finally the Northern objective was shown. In the early years, as well as following the E.P., it has always been about whether the state has the right to cede just because they do not agree with federal beliefs.
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