-Wild foxes killing wild rabbits is okay. -Your domestic cat killing native birds is not okay. As a pet owner you're already providing it with its necessary food. It has no reason to go out killing native wildlife other than fun. Not only are you harming native wildlife, but you're also putting at your cat at risk for parasites and disease or getting into fight with other animals. -A predator/parasite killing your domestic cat is not okay.
Being a predator doesn't mean you can kill whoever you want.