Yes, you can install backsplash on drywall. However, you need to prepare the drywall before installing the backsplash. Here’s how you can do it–
Scrape any loose or peeling paint to make a flat surface
Removing any remaining dust, oil, or grease with TSP or ammonia-based cleanser
If there are any cracks or holes, sand the area with medium-grit sandpaper
Finally, prime the wall and let it dry