If said posting fell under fair use there is little the publication can do, but that’s on purpose! We created fair use laws specifically to allow that behaviour. If that is not what we want anymore, logically we would revert the law, not come up with all kinds of weird and contrived bandaids to help only special friends.

If the post fell short of fair use, the publication would have the legal right to seek penalization for the person who posted the content. It would also be a violation of Facebook’s terms of service. In this case, the answer to your question is essentially yes. It cannot be prevented, per se, but corrective action can be taken – which also serves to dissuade others from doing the same in the future.