Wood can always be glued back in place or replaced. The question is *what* fell off, and how much work is involved in repairing it? Without pictures, we can only guess. The underside of the bridge is reinforced with a wooden piece called a "bridge plate". It really sounds like you damaged this. That would explain why the bridge pin won't stay in.
There are several fixes for this. Probably the *correct* solution is to have a luthier replace the bridge plate. If your guitar is expensive or vintage, this is definitely the way to go. No matter what you decide to do, at least have it looked at. Your local guitar shop can give you a free estimate of repairs, and then you can decide how to proceed.
If you have an inexpensive guitar.....say, under a couple of hundred bucks....it's not going to be worth paying somebody to replace the bridge plate. You might consider installing a "PlateMate".
http://www.mitchelsplatemate.com/platemate
Stewart MacDonald sells tools for building and repairing guitars. They have a tool for everything, including bridge plate repair. Probably not a cost-effective solution for one repair, but their video gives a good overview of how the bridge plate works:
https://www.youtube.com/watch?v=iSm4kMSlIYo
I repeat: without photos, we can only make guesses as to what is actually wrong and how to make the appropriate repairs. Taking it to a guitar tech is the only way to know for sure. I've posted some possible fixes, but this is only to give you an overview of the possibilities. Good luck!