In EVERY war; each side is convinced of this!
The Civil War wasn’t like every other war. The United States in 1861 was still very much a Christian nation. Unlike most of the European wars, it didn’t just involve a professional military it also involved ordinary citizens via the draft. It was also the first modern war, with modern death tolls.
Within the church both the north and south were absolutely convinced of the moral superiority of their side. While politically slavery was a side issue for the north, within the church the abolitionist movement was strong and forceful. Within the southern churches support for slavery was strong, in addition to a number of other grievances the south had against the north.
This isn’t a matter of both sides believing God is on their side and well, every war has this. This matter cut much deeper, especially after 600,000 people died and the south and its culture lay in ruins. To say nothing of slavery ending. The civil war directly led to the progressive movement both within politics and the church. The south, which before the war was largely reformed, presbyterian and anglican, became largely baptist after the war. The south rejected the church and shifted to the social gospel via the baptist movement. The north went in a different direction, keeping the old church institutions but gutting them of all their Christian content.