Thanks Ben, I appreciate the helpful comments. I understand what you saying about simply leaving the existing tile backsplash and applying a new tile backsplash on top of it, but I don't think that would be a wise choice in my particular case for 2 reasons:
1. The primary reason that I want to do this repair is because I have entire 6x4 panels of mosaic tile coming loose and falling off the masonry wall. I don't know if that's because of improper installation when the place was built, settlement of the structure, or previous moisture problems. But whatever the reason, my existing tile is not a good candidate for a tile-over.
2. My granite countertops have a 4-inch granite backsplash, and then ceramic mosaic tile on the wall above that. Although the thickness of the mosaic tile is not quite as thick as the granite backsplash, it is very close. Thus applying another tile backsplash on top of the existing means that the new tile backsplash would actually protrude out beyond the existing granite backsplash - something that I think would look a bit hokey.
I know how difficult it can be to remove ceramic from a wall. In my particular case however, I think the tile will easily pop off with very little coaxing. The other option that I am considering is to simply glue the loose panels back onto the wall, re-grout the joints, and call it a day.