I shut the car door on the release pull of my garage door opener... as I backed out, it pulled out. Can it be fixed?

Yes it can be fixed. I assume the door still goes up and down OK. If the release handle did not break it is an easy fix. On the bottom of the release handle there is a small hole. About the size of you pinkie finger. Simply put the rope back it the hole and tie a knot. Then try it. The rope was to long to start with if it got caught in your car door. It should only hang down about one foot. If the rope broke or the knot came out this should fix you up. If the hole broke on the bottom of your release handle it is another thing. It can still be fixed, but you can check it out before I go into that. Let us know how it turns out.
Mike

I'm not that lucky! The semicircular shaped piece that the rope is tied to, as well as the piece that goes into the piece that slides... that all came off!

Any chance of a picture? From what you tell us so far I think your door opener will no longer operate since you "remover" the catch on the release mechanism. Its semi-permanently released 24/7 as it sit. What brand opener do you own?

Ouch... you will have to replace the piece then. Is your door working? Could you past a picture of the broken part ? The part still attached to the door guide. And what make and model of opener do you have?
Mike