We can distinguish between our present ignorance and something that's in-principle not scientifically explicable, surely. We could obviously not use empirical observations - they depend on the ordinary course of physical processes. The answer is, nothing. Non-natural explanations are not scientific. A final form of naturalism is ontological naturalism.
Many scientists are also physicalists.
They argue that if we do not need to postulate the reality of non-physical processes for science, then we can conclude that there are no such things. This argument is too quick. B. Moreover, many things in the physical world are caused by many things together rather than just a few. So, we might say that a physical event is caused both by God and by the physical causes, without being logically inconsistent. Your resolution depends on what you are using as basic assumptions.