The South disagreed, believing that the Federal fugitive slave law should preempt states' rights.
Thereupon the Southern states seceded in order to form a new federal government of their own, under which states were not free to make their own laws regarding slavery.
So yes! the Civil War originated out of a states' rights issue.
Interesting summary, largely correct, except that we don't really know how actively a few Northern states resisted the Federal government's efforts to enforce the 1850 Fugitive Slave Act.
So far as I can tell, the whole issue of "runaway slaves" was hugely exaggerated by Deep South Fire Eaters to encourage secession.
In fact, there cannot have been more than a rare handful of Deep South slaves who escaped via Underground Railroad to freedom in the North.
Distances were too far, and all along the way waited squads of slave-catchers looking for anybody they could grab and sell back into slavery.
Southern Border States (Delaware, Maryland, Kentucky & Missouri) were a different story, of course.
There slavery was much weaker to begin with, freedom much closer, and many former slaves had already been freed by their white masters.
Point is: those Deep South Fire Eaters who complained the most about Federal non-enforcement of Fugitive Slave laws were the least likely to be adversely affected by it.