It seems the equations given by Norton really are unphysical, so actually it's not a proof on indeterminism, but simply incompleteness - newtonian physics cannot model some really really fucky situations - its incomplete.
The first law and stitching arguments he makes appear to both be flawed. Having non-zero derivatives of force in combination with zero velocity and zero force is perfectly in accordance with Newton's first law. And in his frictionless ball counterexample, his equation is incorrect because it violates Newton's second law, not because two solutions are stitched together.
Lipschitz continuity is required for guaranteed uniqueness of differential equation solutions, and non-uniqueness can appear as nondeterminism or incompleteness.
I think he reaches the right conclusion but his reasoning is flawed.