Can I install stone veneer as a backsplash and under the bar of my kitchen?

My husband really love the look of the stone veneer and want to put it in our kitchen going from the bar and extended into the backsplash of our kitchen. The only thing is I'm pretty sure we can't just slap it on the drywall. How would we go about securing it to the drywall we have now? Do we have to rip out the drywall and install something else? Any information would help thank you! Oh and we would really like to do it ourselves to keep costs down.