We show that if self-similar graph actions satisfy contracting and regular conditions, then the shift maps on the direct limit spaces of self-similar graph actions are positively expansive local homeomorphisms. From this, we obtain that the limit solenoids of self-similar graph actions are Smale spaces and that the stable Ruelle algebras of the limit solenoids are strongly Morita equivalent to the Cuntz-Pimsner algebras constructed from self-similar graph actions by Exel and Pardo. We also compute $K$-theory of the stable Ruelle algebras of the limit solenoids.