First, the Southern states couldn't give it up peacefully. They're wealth and culture depended on it. At least, the wealth and culture owned by the elite slaveholders. But that is not why Lincoln called upon the Northern states to provide troops. Nor was preserving the union a unified cause. You might remember the tenacious opposition fielded by the Peace Democrats, nicknamed the Copperheads. They pressured Lincoln to make peace with the Confederacy.
The most fervent Copperhead criticism of the administration revolved around race. Copperheads were deeply racist, even by the standards of their own time. Their racial fears increased as slaves ran to Union lines in search of freedom. Laborers feared that freedmen would take their jobs. Others simply did not want African Americans anywhere near them. Whatever the case, Peace Democrats were convinced that abolitionists had brought on the war by stirring up so much trouble in the antebellum period. Under the Lincoln administration, abolitionists had far too much influence and were, claimed the Copperheads, running the government. (Never mind that Lincoln was a raging moderate who moved to the left only as the war forced his hand.) Tensions were so high in the summer of 1862 that race riots broke out in Toledo and New York.
When Lincoln issued the Preliminary Emancipation Proclamation in September 1862, the Copperheads were apoplectic—although it did give them the satisfaction of saying “I told you so.” The Emancipation Proclamation confirmed their worst suspicions of what they thought was the true agenda of Lincoln and the Republicans: freeing the slaves. Democrats performed well in the off-year elections six weeks later, but Republicans held their own. The Republicans lost seats in Congress, governorships in New York and New Jersey, and legislatures in Indiana and Lincoln’s home state of Illinois, both of which went to the Copperheads. But Republicans had not lost as much ground as parties in power generally had in congressional elections. --
Lincoln’s Critics: The Copperheads, Jennifer L. Weber, Journal of the Abraham Lincoln Association, Winter 2011 (Long read)
As for the Constitution, almost from the beginning the binding of states to the union was questioned. Before the Civil War settled the matter by force of arms, three earlier threats of secession were waved about with varying resolve. James Madison and Thomas Jefferson arguably formed the states' rights-compact doctrine in the Virginia and Kentucky Resolutions of 1798. New England Federalist toyed with the idea of withdrawing from the Union at the Hartford Convention in 1814. Squabbling over expansion which resulted in the 1820 Missouri Compromise saw secession threats. And in the 1830s South Carolina went to the brink of secession over tariffs.
The Civil War was fought on questionable constitutional grounds, but the preservation of the Union, the Emancipation Proclamation, and the 13th Amendment morally justified the outcome.