The threads on my bike's pedal crank just became fully stripped and the pedal fell off, though the pedal itself is fine. Is there any way I can fix this without fully replacing the pedal crank?
I have seen people succesfully filling the the whole space of the crank thread with weld, then re-manufacturing the thread in a turning/lathing shop.
I have done this to a bent/stripped derailer hanger of a steel frame, and it worked perfectly.
I think you could weight price, availability and safety of each alternative, but very probable that replacing the crank is the wisest thing to do, unfortunately.
It might not be so in case of a very expensive (and high quality) crank, but if the crank is medium or low priced, replacing is the way to go.
If the crank arm is wide enough it may be drilled and a thread repair insert installed or tapped to a larger size. This is best left to your local bike shop. To buy the tools that you will most likely only use once will exceed the labor fee they will charge. They can also check to make sure the repair can be done safely.
This happened on my Trek Pilot. The pedal was a bit loose and I didn't fix it in time before it bore itself out of its hole.
My LBS drilled out the threads and installed a new helicoil. It's a simple task, I think I paid less than $30 for it (although I did get a friend-of-the-shop discount, you might pay more).
Park Tools has instructions on how to drill and then tap a new bushing: http://www.parktool.com/blog/repair-help/pedal-bushing-repair-kit-procedure