One reason that South Carolina was not justified in leaving the Union was constitutional. In short, South Carolina, along with twelve other states, had chosen to join the Union when it ratified the Constitution in May of 1788. Like every other state that ratified, they joined without qualifications or conditions. The Constitution contained no provision allowing states to leave the Union—indeed, it would have been self-defeating to include such a stipulation. So there was no constitutional basis for secession. To do so was to essentially nullify or negate the Constitution.
South Carolina, in announcing its decision to leave the Union, claimed it was justified in doing so because several Northern states were violating the Fugitive Slave Act (which itself was based on a clause in the Constitution), yet South Carolina itself had claimed the right of states to nullify laws in 1833, when it provoked a crisis over nullifying a tariff its leaders saw as injurious to their interests.
Finally—and most importantly—to argue that South Carolina was justified in leaving the Union is essentially to argue that the state had the right to continue to enslave human beings. In a "Declaration of the Immediate Causes of Secession" issued by the state's secession convention in December of 1860, South Carolina complained that Northern states, with the election of Abraham Lincoln, had taken up a uniform and explicit anti-slavery position, one which threatened their rights as slaveholders:
A geographical line has been drawn across the Union, and all the States north of that line have united in the election of a man to the high office of President of the United States, whose opinions and purposes are hostile to slavery.
In short, as the state's leaders themselves said at the time, South Carolina left the Union to preserve slavery. There can be no moral justification for such a cause.