I would argue the opposite--the Civil War was the natural result of the Constitution:
"We are cautioned . . . against faction and turbulence. I acknowledge that licentiousness is dangerous, and that it ought to be provided against. I acknowledge, also, the new form of government may effectually prevent it. Yet there is another thing it will as effectually do- -it will oppress and ruin the people." - Patrick Henry
" In the most limited governments, what wranglings, animosities, factions, partiality, and all other evils that tend to embroil a nation and weaken a state, are constantly practised by legislators. What then may we expect if the new constitution be adopted as it now stands? The great will struggle for power, honor and wealth; the poor become a prey to avarice, insolence and oppression. And while some are studying to supplant their neighbors, and others striving to keep their stations, one villain will wink at the oppression of another, the people be fleeced, and the public business neglected. From despotism and tyranny good Lord deliver us." -Philanthropos (Antifederalist Paper #7)
The Constitution created the national government that waged the Civil War. Before the Constitution, were were a confederation of independent states. After the Constitution, we were a consolidated republic with supreme power OVER the states. The Constitution didn't die in the Civil War. It brought about the Civil War. Like most big government boondoggles, the Constitution brought on the very thing it was supposed to prevent.
Actually, I think we're both right. From 1789 until the Civil War, we were still a "grouping" (by whatever name applies) of sovereign states, operating under the Constitution. The Civil War destroyed that point of view (illegally, in my view). On the moral question of slavery, the South was wrong. On the political question of secession and state sovereignty, the South was right.