My a/c system kicks on and off like it should, but the fan on the outside unit won't spin.

Im not for sure how your system is cycling and cooling with out the condenser fan running unless you are in a real cool area trying to run the a/c. If it works in warm temps, I want what you have to save on my electric bill.(You may have a low ambient kit on the system to keep the pressure up with cool outside temps)

You should take oil the fan if that won't work you will have to replace it

Could be the fan capacitor/motor as well as the other suggestions.