Thanks for the people from this thread for finding it out (I just tried it, and it works, but it might be difficult to discover if you're not really into permissions etc.).
If you have this problem, go to the folder /Applications/Games/Call\ of\ Duty\ 4/Call\ of\ Duty\ 4\ Data/zone/english. If you do this in the Finder, you will see a few black icons there. If you do "Get Info", you'll see that they cannot be read by everyone, and that your account is missing from the list. Change it so that everyone can read them and you're done. You can also do this from the terminal. Go to the same directory, make yourself administrator and type "sudo chmod +rx *" (or just the names of the black files).
I think the problem is that we installed the game under our own account, using drag&drop, and then ran the updater as administrator. That makes that the new files are inaccessible. The above procedure fixes that.