How do we deal with objectionable content already on the internet? It gets taken down by law enforcement of some country that finds it objectionable, and we handle the case in courts.

A new protocol that makes this content more accessible isn't an issue with the protocol, but with society and how we decide to deal with it. If CP is found to be served by nginx over HTTP, is that a problem with nginx or HTTP?

If anything, centralized services only make the problem more difficult to address, since they're expected to serve the demands of governments, companies and law enforcement agencies worldwide, while somehow being the arbiter of free speech. Those are impossible goals to reach, and go against the original design of the internet.

This discussion is as old as P2P protocols. I'm sure that if Nostr became a popular way to share copyrighted content, governments would try to fight it, just as they've done before. But at the same time, censorship is not something a protocol should care about, and just like BitTorrent thrives today, with enough interest, Nostr would also find a way to persist.

In many European jurisdictions, you are criminally liable for anything that makes it onto your systems, regardless of how it got there.

So, the OP has just given people a whole new way of making themselves liable for illegal content, just by running the P2P software in question.

In most cases, the police aren't interested in hauling a service provider into jail, they want to haul in the person who put that content there. But this does make a nice little lever that the police can use against any service provider.