Why did the Union and the Confederacy believe the American Civil War was justified?

The North believed that war was the only available option to keep the Union together. The South had seceded and fired the first shots at Fort Sumter. This was an act of war, and the appropriate response was a similar show of force. Though the nature of the Civil War changed considerably over the course of its duration, for the vast majority of Northerners, its aim remained the same: to prevent the country from splitting in two. The integrity of the Union was at stake, and with President Lincoln in charge, most Northerners believed that it was necessary to fight for it.
In the South, the general consensus was that the Civil War was necessary to protect states's rights (namely, the right to maintain the system of slavery). Ever since Lincoln was elected President in 1860, the Southern states had begun to secede from the Union, fearing that slavery, the very cornerstone of Southern economic and social life, was under serious threat. Though Lincoln had explicitly denied that he would take steps to abolish slavery if elected, the South simply didn't trust him. Even if Lincoln didn't abolish slavery outright, the South feared, at the very least its spread would be contained, thus diminishing the power of the Southern states within the American political system. It was thus felt in the South that the only way to secure the newly established Confederacy, and by extension the institution of slavery on which it was based, was through armed conflict.