Why don't airliners have a navigation GPS as back up in case of a pitot tube malfunction?
Airliners do generally have GPS receivers, but GPS doesn't give you the information that's needed in this case.

In AF 447, the primary consequence of the blocked pitot tubes was that the flight envelope protection system couldn't do its job properly and therefore automatically disengaged together with the autopilot. The subsequent crash was caused by the pilots' confusion and failure to realize which consequences this had for the aircraft's reaction to their control inputs.

However, what this system needs to function is not something a GPS system can possibly provide. GPS knows how fast the aircraft is moving relative to the earth, but for flight envelope protection you need to know how fast the aircraft is moving relative to the surrounding air, which GPS doesn't have any way of knowing. The differences between these values is caused by wind and can be pretty large. It would be a recipe for disaster to let the computer limit what the pilot can do based on an arbitrary assumption that there's no wind. (And, as Terry comments, there's an additional difference caused by air pressure).

If I remember correctly, the AF 447 investigation concluded that the pitot tubes had thawed and began functioning properly well before impact; it was not the loss of airspeed indication that prevented the pilots from salvaging the situation. By this time the pilots were confused and thought the airspeed indication was still malfunctioning because it showed a much slower speed than they thought they ought to have -- whereas in fact it was working perfectly well.