What is the minimum distance wiring must be behind a tile backsplash?
Your issue is covered in the National Electric Code, for installations in the USA:


  NEC 300.4 Cables and raceways shall be protected from damage. Where installed through bored holes in wood framing members, the holes shall be bored so that the edge of the hole is not less than 1¼ inch from the nearest edge of the wood 
  member, or shall be protected by a 1/16 inch steel plate.


So basically you need 1¼ of solid wood material (not counting the wallboard/wall covering) to buffer access to the wire, or a metal plate.

But in your case I'd I recommend using armored cable (formerly known as 'bx').  Or emt (conduit). Then you've got no worries.

Frequent practice is to have the fridge on it's own circuit, and the dishwasher (if you've got one) is required to be on its own.