Can I install Slate Ledger stone panels over drywall?
I would be concerned about putting that much over all weight on drywall. Even though it may be supported by the floor. The facing on drywall is only paper. When the thinset is applied, the paper will get wet, which it will, even with coats of paint over it. Even if the paper stayed bonded to the gypsum core, that is all it is bonded to. Gypsum has no real strength, not for that kind of shear forces. And on top of that, it may have a chance if the drywall is glued to the studs, but if it isn't, the few screws or nails is all that is keeping all that weight from buckling the drywall and falling.

In my humble opinion, use 1/2" cement board.