Most people’s political convictions resemble those of Captain Renault from the classic film Casablanca; it blows with the wind, and the prevailing wind happens to be not entirely hostile to states’ rights.
The victors of war write its history in order to cast themselves in the most favorable light. That explains the considerable historical ignorance about our war of 1861 and panic over the Confederate flag. To create better understanding, we have to start a bit before the 1787 Constitutional Convention in Philadelphia.
The short answer to your question is that the Civil War had no effect on state sovereignty and nullification, at least not legally. Some people may point to the Fourteenth Amendment as evidence that the federal government acquired expansive new powers over the states, but that argument doesn’t hold weight when scrutinized closely.