Yes. It is clearly defined in Hawaii state law.

Yes, all beaches and coastal areas to the high tide mark are public property according to the state constitution. Pretty much all coastal areas are already owned by the federal or state government and lava adding to it would just add to that.

Aren't there privately owned beaches in Hawaii?

Nope. You might have trouble getting there though. If there's no public road/easement to the beach, you'd have to get there by boat in order to legally use the beach.

And the line above the high tide line is where it ‘stops’, so you’d need to leave or trespass pretty quickly.

You can own property abutting a beach, all the way down to the mean high tide line. Below that that, it's public.

Not even Mark Zuckerberg can own a beach in Hawaii.

For a few billion I bet we’d sell him one of the small uninhabited ones.

There are over a hundred, ranging from tiny to a few thousand acres.

I'd sell him the stretch of beach that is most highly active with lava flows.