Masur and Minsky, in their paper "Quasiconvexity in the curve complex", give a purely combinatorial proof that certain "nested" train track sequences project to quasigeodesics in the curve complex. It follows from this, without too much more trouble, that a pseudo-Anosov mapping class has positive stable translation length. Teichmuller theory is not used in this proof.

-----edited after Dave Futar's comment-----

Here's an argument for the converse, following Dave's suggestion to use Klarreich's boundary. Suppose that $\phi \in MCG(S)$ has positive translation length. So $\phi$ has a unique attracting/repelling pair of fixed points on the Gromov boundary of the curve complex. By Klarreich's theorem, this is a pair of measureable but measureless singular foliations $F_1,F_2$, each of which is arational. They are unequal in the Gromov boundary, so by Klarreich's theorem they are inequivalent (with respect to isotopy and Whitehead moves), and so they can be realized on the surface as a transverse pair of foliations. And there is an actual homeomorphism $\Phi$ in the mapping class $\phi$ that preserves this pair of foliations. What's left is to construct transverse measures on these two foliations with the appropriate stretching properties. This situation occurs as a piece of the proof of Thurston's classification theorem in Fathi-Laudenbach-Poenaru, and the conclusion is that either $\Phi$ is a finite order mapping class or the desired transverse measures exist and $\Phi$ is a pseudo-Anosov homeomorphism. Finite order is ruled out by the assumption of positive translation length in the curve complex.