This sounds wrong to me - Ohm's laws don't apply to all materials, but Newton's laws don't apply at all speeds; what makes one more "fundamental" than the other? Is it just that it took longer for people to find exceptions?

I think he had in mind the fact that Ohm's law can be derived from a description of the behavior of charge carriers and their interaction with the material, which are basically an application of Mazwell's equations, while Newton's laws are not explained from other ("more basic") principles. In the case of Newton's laws, you either start from them, or replace them with a new description of spacetime itself.

Ohms law is an empirical statement, like hookes law, "the current density is proportional to the applied force". Since the applied force is an electric voltage:

J = s*E where s is the conductivity, J is the current density and E is the electric field. This is the real ohms law, from which it is possible to derive V =IR.

If you think about it, it is surprising that ohm's law should ever hold. E accelerates charges, so as they accelerate, the current should go up! Then V = IR would depend on how long V has been running.

The solution of course (because ohm's empirical law fits the data) is that changes in the wire propagate so fast, the current quickly settles in to a steady state due to collisions between electrons balancing the net acceleration due to the E field.

This is adapted from Grifith's Electrodynamics, which is a great book. He explains it better than I do.

Anyway, it should be easy now to see why Newtons laws are more fundamental (not absolutely fundamental of course) than Ohms Law.

Not really. The point of Ohm's Law is that in a resistive medium a charge carrier (electron) experiences so many collisions that it very quickly achieves terminal velocity (aka drift velocity). Essentially, the current density will be a function of the applied electric field [j = j(E)] and, to lowest order, the current density varies directly with E with the proportionality constant being defined as the conductivity in an isotropic medium (it's a tensor in anisotropic media).