My engine began overheating so I began a process of elimination for the problem, only to find that my radiator cooling fan had fallen off the motor that turns it and is now just resting inside the fan's plastic inner cover. Would fixing this be as simple as reattaching the fan? Or is this a serious problem? Thanks for your help!

 

I would suggest getting another set from the junk yard or Craigslist.
Should be cheap and easy to find.
As far as reattaching the fan, I wouldn't - next time it comes off it may puncture the radiator costing you more $$ to fix.

 

Fan probably got "reamed out" by the fan shaft and a loose nut. It happens. A new fan assembly is not too expensive and these motors are not repairable. Check ebay, amazon, rockauto, et al for aftermarket fan assembly.

good luck