State's right to do what?
The right to secede. A right not prohibited in the USC.
Own slaves.
The South wanted to expand slavery to the territories, the North didn't.
If Southern Democrats believed so much in 'states rights' they wouldn't have pushed the 1850 Fugitive Slave Act through Congress.
The states' rights to LEAVE. And this is why some people think you are deliberately obtuse and intent on remaining that way.
I suspect you were fishing for the answer "keep slavery", but that is just misdirection. The states already had the right to keep slaves, and they would continue to keep having the right to keep slaves indefinitely so long as they were part of the Union of the States.