Consciousness is defined as, roughly, conscious personal experience. Nervous system-bearing organisms have it and tables don't...In theory, we could all be as mindless and devoid of consciousness as tables and from an outside observer's POV, nothing would change in our lives, our speech or all of human history.
If you can separate consciousness from the nervous system, as you suggest in theory, then why can't a table have consciousness? Table consciousness would be very different from human consciousness, without the response to sensory input that a nervous system provides, but it could have a sense of 'be'-ing even if that leaves out any awareness of anything else.
More to the point, it seems there's no reason to dismiss the possibility that matter/energy itself is conscious in some way. That, indeed, consciousness is inherent to everything in the Universe, including the singular consciousness of the Universe itself. Why not? How can this be falsified?
As to proof, that's difficult if not impossible with the collective cultural bias against a self-aware Universe. I'm not taking a stand one way or the other, but it seems a glaring omission of scientific research to not approach some of our observations of the quantum and galactic worlds with the reverse assumption that the Universe is in fact conscious. Where would that lead the research? Could research be forked with this assumption in mind?
On a somewhat related tangent, in my experience there is a kind of proof that anyone can tap into that is greater than any proof offered by science, society or religion: direct inner inquiry. While we need to temporarily accept certain things as truth to get by, it seems foolish to fully accept anyone else's point of view as truth unless it is vetted by one's own inner inquiry. Meditation is a great tool for this -- learning to simply observe your own mind, body, breath and emotions. You can be astounded by what you can learn by observing your own consciousness. Matched with the objective evidence of your outer world and more than a small dose of patience, you can go far in life. This is the result of "spiritual or mystical revelation" -- revelation from within yourself, resounding in truth that you prove to yourself through action and results. This is a completely different experience from a belief that is taught from "out there" -- be it religion, TV, God, or Science (to some, those are all the same thing). Those things may teach you truth, but it is only actually truth after it is vetted by your inner guidance.