To be pedantic - that would only show that consciousness is non-neuronal. Max Tegmark has this idea of "consciousness as a state of matter"[1] - that neurons aren't particularly special - what matters is are particular patterns. These consciousness patterns can be composed of neurons, but they could also be composed of computer circuits, or rain drops, or any other particles.[2]

To be pragmatic - the absence of a physical cause for consciousness does not preclude any other causes. Whatever environment consciousness exists in that environment must have rules, laws, constants, and symmetries that define that environment. So all we're doing is trading the laws of the universe for the laws of some ethereal no-material realm or supernatural realm. We're not actually performing any useful work, we're just shuffling the problem around.

The hard problem of consciousness, the questions of determinism and free will - these problems will challenge any consciousness that finds itself in a rule-bound system.

[1] https://arxiv.org/abs/1401.1219

[2] https://www.youtube.com/watch?v=GzCvlFRISIM

Tegmark's theory sounds similar to Chalmers, yet Chalmers calls himself a dualist.

Substituting a few words to get to a less polarizing and more obvious topic:

If falling isn't the result of gravity, then that would be a good reason to suspect that physics is somehow false.

If it's not neurons, you're going to need extraordinary evidence!

If it's not neurons, it may be that the foundation for understanding consciousness is wrong. Which would be assuming that some objective, abstracted view of the world can explain our subjective experiences.

I'm not saying it's impossible. Who knows. But it remains a hard problem for now, at least.