If you found something that could solve the halting problem and you're lucky enough to be able to prove it for this special case, that would just mean that the assumption that everything physical can be computed by a Turing machine is wrong. It works the same way with anything else that we think of as "impossible". If you observed an apple falling towards the sky that doesn't mean that the apple is non-physical, it just points to a serious flaw in our understanding of gravity. Experiments trump all physical theories.
Technically, we already recognize the world is non-physical. Materialists defined materialism to things bumping into each other. Now we know there are things like field effects, attraction, and action at a distance. So there is a non-bumping substrate that the bumping things exist within. Thus, strict physicalism is already known to be false.
The term physicalism was introduced a few decades after we figured out Maxwell's equations. If the philosophers defined it to mean things bumping into other things they were frankly poorly informed about the world we live in.
So, in general the non-physical is defined negatively. The physical is defined by physical limitations, i.e. the laws of physics and computation, and thus anything that surpasses those limitations is non-physical.