Nice paper, I was not aware of it. There is this paper:
I.G. Nikolaev, A metric characterization of riemannian spaces, Siberian Advances in Mathematics, 1999, v. 9, N4, 1-58
see the mathscinet page:
Citing from the Abstract: "We present a metric characterization of Riemannian spaces: a locally compact metric space with intrinsic metric in which geodesics are locally extendable, and which has Holder-continuous curvature as a metric space, is isometric with a C^2-Riemannian manifold."
Edit [by J.O'Rourke]. Following Willie's advice (in the comments), here is the MathSciNet link.