Stone veneers rely on the continuous 1-inch air space just to their interior. With this excellent drying mechanism, they are as green as they are durable. Without it, pale green is a stretch.
In general, exterior above-grade walls should have continuous air, water, and thermal barriers.
Installing rigid insulation as the exterior sheathing can accomplish all of these tasks, but you need to account for shear resistance (provide wall bracing) in other ways, and you must provide special details if you expect the foam sheathing to perform as a water-resistive barrier (WRB).
A double-stud wall insulated with dense-packed cellulose is an excellent thermal barrier; however, double-stud walls still need at least one additional air barrier to limit air leakage through the cellulose.
Check out the advanced framing section of Construction Detail Library for alternative techniques to achieve shear resistance, and check out Using Rigid Foam As a Water-Resistive Barrier for more information on walls without housewrap.
For more information on air barriers, see Questions and Answers About Air Barriers