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I don’t think Nozick's condition is strong enough. Here's why.

Suppose that someone, call him A, appropriates a valuable natural resource X in a poor country. It might be oil or diamonds, for example. Now, according to Nozick's criterion, A can take the diamonds and make any amount of money off them and everything is morally unobjectionable as long as the other people in the country are not made worse off. If he means "materially" worse off, and I think he does, then as long as everyone either stays the same or gains a single penny there is no problem with A taking resource X, making it his own, and making a fortune. The distribution of benefits from the exploitation of X doesn't matter at all.

As a utilitarian, that makes no sense to me. It seems to me that by sharing the benefits of the resource we can increase the total happiness or the average standard of living enormously. That's exactly what we should do.

But even if we are not utilitarians, the people of our hypothetical country have lost the right to control a valuable resource in a collective way for the benefit of everyone (or at least for the benefit of most people). They might want to nationalize the resource, but that doesn't matter. Losing the option to do that doesn't seem to count as being "worse off" to Nozick. But it seems to me that the loss is enormous.

It seems to me that if we are to justify the original acquisition of property in limited natural resources, there must be a way to compensate the people who lose access to those resources because others have acquired them. In other words, people can only acquire private property in limited natural resources if they do so within a framework of rules that requires that substantial benefits go to those who lose access to those resources. If you want a non-utilitarian justification for regulating property for the good of the majority, I think that's it.