The debate over federalism did not contribute to the Civil War, it was the cause of the Civil War. Sectional differences between the north and south had existed long before the Civil War however, when Missouri territory applied for statehood the Congressional balance of power was threatened. In 1820 there were eleven free states and eleven slave states, Congress was at an impass as to the fate of Missouri...should it be free or slave??? Obviously if the balance of power was disrupted one side would be at a disadvantage. In addition, southern state governments argued that the 10th Amendment guaranteed the property rights of slaveowners since the Constitution possessed no authority over what kind of property could be owned. On the other hand, there were those who argued that Article VI of the Constitution had bearing upon the situation. If slavery were abolished on the federal level the state governments would have to submit. By 1860 communication between the north and south had completely broken down leaving only the last resort...war. This war forever settled the tug of war between the state government and the federal government with regard to division of power. State governments must adhere to the Federal government. The evidence can be found in our written words...before 1865 the nation was always referred to as 'the United States are' after 1865 the nation was referred to as 'the United States is'.
The question of the autonomy of the states as opposed to the federal government's power was reflected in both sides' beliefs in the start of the Civil War. The federal government as being supreme thought process was seen in the North with its professed faith in the federal government. The argument for state sovereignty and autonomy was best represented in the South, which argued that the issue of slavery was one of local practice and represented a domain where the federal government had no right to intrude. Southern states' acts of secession from the Union was a response geared out of the desire to maintain state freedom and autonomy in the face of an encroaching power.
The question of federalism and states' rights played a role in the civil war, but the bottom line is that most southerners didn't benefit from the slavery system. It was mostly the well off plantatation owners who felt they couldn't survive without free and inexpensive labor, so the answer to laws that would free the slaves was to secede so that they could continue the ownership of free labor.
Remember that the Civil War was essentially about secession. The southern states left the Union and the northern states remaining fought to restore it. As historian Shelby Foote argues, the southern states never would have joined the country in the beginning if they didn't think they could leave. Therefore they believed it was a state's right to secede and the federal government could not stop them.
The fact that Lincoln had been elected in 1860 without a single southern electoral vote (his name wasn't on the ballot there) was the last straw, as they felt his abolitionist tendencies would trample on their entire way of life regarding slavery. So this was a war caused by the states vs. the federal government over issues of power.
You can say that the Civil War was about nothing but federalism. The Southern states wanted to be able to do their own thing when it came to slavery, without any interference from the Northern states. The Southern states were unhappy with the idea that the federal government could make any sorts of regulations about where slavery could and could not exist.
The South was also generally concerned about the federal government being able to make other sorts of laws that impacted the South. This was seen most clearly in the nullification crisis over the tariff of 1828. The South felt that the tariff hurt it and helped the North. The South wanted the power to nullify laws like this -- that means they wanted more states' rights and less federal power.