ACP:
IMHO, I think any natural stone could not be cut by normal scroll saw blades. The Teeth of our wood and plastic blades just won't cut natural stone. Glass cutting diamond friction cutting blades are available, but gosh they are expensive. This link:
http://www.bgartforms.com/Artisan_Scroll_Saw_Blades.htm scroll to bottom of page.
Note: friction cutting blades require a cooling lubricant drip system, which I believe only the high end Hawk scroll saw has a system adding a drip system. You would also need a scroll saw whose lower arm and blade chuck are designed for a drip system (water and rust and all that.) See the following:
http://www.hawkwoodworkingtools.com/ and look for accessories - scroll saw. Hawk industries may sell diamond friction cutting blades also.
A Hawk scroll saw will not be inexpensive.
But I am asking, have you considered doing a Google search on a PC driven Water Jet cutter? Or even a sandblasting cabinet? I would think, not having tried it myself, that a PC driven water jet cutter would price out about equal to a Hawk and accessories.
BTW: Welcome to the forum. I hope you get the advise you are seeking. Others, I am sure, will add their advise.
Phil