Does the FAA have risk targets for specific activities?
"We rate passenger airline flights as 'safe' if there are fewer than X casualties per million flying hours"

I am not aware of any official FAA document that states such sentence. However, something similar can be deduced from safety requirements related to the development of aviation systems. 

ARP4761 is a well-known standard for conducting safety assessments on civil airborne systems and equipment. A probability of failure occurrence associated with the severity classification can be found on page 14, which shows the following table:



As it can be seen, the maximum allowed occurrence of a catastrophic failure is 1.0E-9. Playing a bit with the numbers, this could be translated in:


  Commercial aviation flights are considered as 'safe' if there are fewer than 1 catastrophic condition per billion flight hours


One could assume that a catastrophic condition can lead to fatalities, and this seems reasonable. But I'm afraid that this presumption is not written in official FAA publications (I might be wrong).



P.D.: Now, where these probabilities of occurrence come from is another interesting (and un-official) story, but I think is out of the scope of this question for the moment.