There were many things that combined to cause the Civil War, but the root of the entire thing was slavery. The "state's rights" argument is whitewashing the central issue, which is that states seceded to preserve the institution of slavery. But don't take my word for it, just take a look at the Texas declaration of secession:

In all the non-slave-holding States, in violation of that good faith and comity which should exist between entirely distinct nations, the people have formed themselves into a great sectional party, now strong enough in numbers to control the affairs of each of those States, based upon the unnatural feeling of hostility to these Southern States and their beneficent and patriarchal system of African slavery, proclaiming the debasing doctrine of the equality of all men, irrespective of race or color--a doctrine at war with nature, in opposition to the experience of mankind, and in violation of the plainest revelations of the Divine Law. They demand the abolition of negro slavery throughout the confederacy, the recognition of political equality between the white and the negro races, and avow their determination to press on their crusade against us, so long as a negro slave remains in these States.

(emphasis mine)

There's plenty more that can be said about it, but our FAQ is a great place to start.

There's always more that can be written, but I've delved pretty deeply into the state's rights lie in this older answer. The very short version is "none whatsoever because they all come back to slavery almost instantaneously, when given a second's scrutiny."