You haven't shown that it's even possible to observe the supernatural scientifically. You're just stating that it is over and over again without explaining how you might do so.
You don't have to, and in fact it would be absurd if that was a requirement. If the supernatural doesn't exist, then it's impossible to observe it. You can't devise a test to observe the impossible or non-existent.
Going back to the gravity scenario, can you devise a test to show gravity isn't real? I can't. You can test gravity, and I can tell you what observations would disprove the idea of gravity (namely we'd have to observe that matter isn't attracted to other matter). If naturalism is not correct, then we'd have to observe something that is outside the natural world to disprove it.
If personal experience counts as observation, then naturalism has already been disproven. Any aspect of alleged supernatural phenomena that can be studied through the scientific method is not itself supernatural. We can study physical aspects of NDEs, for example, but how can we observe whether or not they're objectively real?
Even if we were to witness a bona fide miracle, it would not disprove naturalism, because the naturalists could say that some day, we might have the means to explain what occurred scientifically, even if we currently do not. Any sufficiently advanced technology is indistinguishable from magic, and vice versa. This is the error of ID and their attempts to prove design.
Naturalism (or anything else) isn't categorically disproven by someone's anecdotal observation. You must confirm the observation.
And I agree, it's probably true that science can't fully study the supernatural if the supernatural is real. We'd have to devise some other method to test for the supernatural. However, people who believe in the supernatural still would have no justification to do so until we figure out that test. We'd be left with an unexplained phenomenon, and that's it.
Naturalism is more of a response to supernaturalism than anything though. Without people believing the supernatural, naturalism is simply the default. There wouldn't be any reason to even have the concept of naturalism as a worldview, even though everyone would effectively be naturalists.
As it stands right now, we have no reason to believe any other worldview other than naturalism. If we find reason to disprove naturalism someday, then that means we are currently wrong. That doesn't change the evidence and justifications we currently have, and it doesn't mean that we will one day disprove naturalism. Based on current evidence, naturalism is the only rationally justifiable position to have.
Naturalism is a metaphysical concept, just like materialism. Materialism has been demonstrated to be a moving target--it was disproven by electromagnetism, but shifted its definition slightly to accomodate it. Naturalism can and, if necessary, will do the same.
These are not scientific theories. They're metaphysical constructs, and they can be stretched as necessary to account for new information. Anything that is too vague to even be defined properly is not falsifiable.
The flaw in your argument is that nobody is arguing that we know everything there is to know about the natural world. If we discover something new about the natural world that we didn't know before, that doesn't disprove naturalism, it would be incorporated into it.
If ghosts turn out to be real, however are shown to be completely naturally occurring, that would be surprising however it's not inconsistent with naturalism. It would be an aspect of the natural world that we were previously unaware of.
If anything, discovering naturally occurring ghosts would be a strike against supernaturalism, if one of the main things cited to be supernatural was actually discovered, and wound up not being supernatural, that goes against the idea of the supernatural being real, not the natural. It only means we previously put that idea in the wrong category.
If we discovered supernaturally occurring ghosts however, then you have a clear proof that naturalism is false.
This is metaphysics, not physics, so it's not really up to anyone to prove anything. Just to defend your own position and point out the weaknesses in your oponent's.
"Only the natural world exists" is not a more neutral claim than "The natural world is not all that exists." I would say that the former is actually the more radical of the two ideas.
Occam's razor disagrees with you. We have no reason whatsoever to believe in the supernatural, and we have no reason to believe its existence is required for anything. We know the natural exists, and we know the natural works. You are needlessly and unjustifiably bringing a whole extra realm of existence into the equation for no reason whatsoever.
And hey, perhaps it's actually real, however as it stands we have no reason to take the idea seriously. Come back with better reasons, and then it's worth further consideration.