Would a proof assuming a physical law be considered sufficient?
It seems at least possible to me, but currently very unlikely. To sum up the below, it's because the current mathematical statement of (say) P vs NP is completely independent of any laws of physics, so one would need to describe new models of computation that do depend on physics axioms.

The key point that Peter Shor made in his comment is that CS theory questions, such as P vs NP, refer to very simple and stylized mathematical models. They aren't statements about the real world. They just say "in this mathematical model, ___ is true".

Now, one does often have empirical laws, such as the Church-Turing thesis, that state that the real world acts like these mathematical models. But that's a one-way connection (it doesn't mean that mathematical models must act like the real world!). To flesh out Peter Shor's example, the Pythagorean theorem only needs the ideas of real numbers and the Euclidean plane/distance. The model is much simpler than the real world and doesn't involve e.g. gravity, electromagnetism, thermodynamics, etc. And even if the Pythagorean theorem were sometimes false in reality because of these complications, this would not affect its mathematical truth.

Similarly, the model for the Turing machine and the definitions of P, NP, etc are much simpler than the real world. The model does not involve things like gravity, thermodynamic entropy, etc. The truth of P vs. NP does not depend on whether computation can actually happen efficiently in the real world.

Now, it seems to me hypothetically possible that in the future, we could discover closer connections between laws of physics and laws of computation. What would happen then is that the mathematical model of the Turing Machine would have to be expanded to account for these connections. One would then have to formulate new definitions of P and NP for this new model and argue that these are "better" than the old model and definitions. Then, in this new physics-aware model, one could have physics axioms that are used in proofs. But this seems very unlikely / far from happening, at least to me.