In my experience it usually will but not always, I have one that two thirds has grown back OK but the other third does it's own thing, grows and then breaks off again. All good fun,should be older enough to know when to keep my fingers out of the way by now.
Wow, that really had to hurt! Nails will usually grow back if they have ripped off or were removed. However, the nail probably won't look the same as it used to, and it usually takes several months for it to completely grow back.
Please keep the injury bandaged and watch out for any signs of infection.