Did the states vote on the issue? Either way I cannot reconcile saying no to them not being allowed to secede with Wilsons 14 points (and, on a personal point, I very much enjoyed my childhood in Southern Jutland, which was one of the areas that voted to join another country after WWI) or even the American revolution. I also can't argue against going to war with them to end slavery but that shouldn't necessarily mean that they should be forced to join the union afterwards.
To my knowledge, states did not hold plebicites. However, their legislatures did vote on the issue. You can read the ordinances of secession passed by the legislatures here: http://www.civil-war.net/pages/ordinances_secession.asp
Note that what is shown are excerpts. click the links to view the full measures passed. Note the ones that include the defense of the institution of slavery as their primary motivation.
Secession had been a topic of politics in the South for decades, and was was unpopular for a long time. After the South Carolina Fire-Eaters forced the issue in 1861, a wave of secession fever swept the South. Most seceding legislatures passed secession with large majorities, except for edge cases like Kentucky (where the state government rejected secession then basically collapsed).
None of this was democratic, of course, because many Southern states had large black majorities. IIRC, More than 80% of Mississippi and 65% of South Carolina, by population, was black. Nearly all those people were enslaved and/or disenfranchised. These were not nations of the people, but states run by and for the slaveowners. The only reason that contemporary Southern states had a majority-white population is because the wave of state terrorism and terroristic violence caused a massive exodus of black Southerners, the Great Migration.
Of course it was democratic. It's easy to forget that the definition of who is or isn't a citizen and who is or isn't eligible to vote is historically impermanent.
It's not even comparable to women's suffrage: women were citizens who couldn't vote because of their gender, whereas slaves were in an entirely different category. The closest equivalent in the modern US would be prisoners (except their status is not intrinsic but temporary). In terms of human rights, slaves weren't really treated as human. Slaves are property, not unlike cattle and livestock.
This mindset is obviously appalling from a human rights point of view, but it's misleading to call a state undemocratic just because it doesn't categorize a large segment of its population as people (the "demos" in "democratic"), even if we wouldn't consider it democratic by our (mostly superior) modern ideas of who should or shouldn't be represented in a representative democracy.
The secessions happened around a time when those definitions began to change and more non-slaves became increasingly uncomfortable with the dehumanizing idea of slave ownership (in part carried by new religious ideas though it would be naive to thank religion for abolishing slavery). One of the reasons this change of mind was met less favourably in the seceding states was that they were the ones most economically dependent on slave ownership and therefore most threatened by any legislation that would make the practice illegal or less viable.
Heck, considering how latent racism still is in the modern US, it should be easy to imagine how the idea that black people are no different from white people except for irrelevant details like the colour of their skin wasn't actually widely accepted at the time. It's not like free slaves were treated as equals to white men either -- although at least recognizing them as human (and citizens) certainly was an improvement.
