I don't think it would go down that way. There were certainly some outside of SC that might be more radically anti-federalist than Jackson but not enough for revolution. My guess is that Jackson pacifies South Carolina with his typical shock and awe. It likely becomes controversial, perhaps destroying the Democratic Party.... But no civil war.
Well, technically, a SC only secession WOULD be a civil war - just a short one.
And, yes, it would establish firmly the principal that no state (or states) could unilaterally secede.
Which would make the slavery issue around the time of OTL's ACW really, really messy.
OT3H, if the more populous north doesn't have to worry about the South going its own way, they MIGHT not be so willing to compromise on e.g. the Missouri compromise.
OT4H.... The South had enough votes to block legislation, and did. You might find that the Federal government becomes relatively ineffectual, as the North won't pass anything the South wants, and vice versa.
So. E.g. No transcontinental railroad, at least nothing like as early.
OR... Maybe two railroads - one south (on the Gadsden purchase), one north (e.g. Great Northern). But in that case, it would be a decade or more later before the money and compromises could be worked out, no doubt.