Vision-language models (VLMs) have shown a promising ability in image geolocation, but they still lack structured geographic reasoning and the capacity for autonomous self-evolution. Existing methods predominantly rely on implicit parametric memory, which often exploits outdated knowledge and generates hallucinated reasoning. Furthermore, current inference is a "one-off" process, lacking the feedback loops necessary for self-evolution based on reasoning outcomes. To address these issues, we propose GeoSkill, a training-free framework based on an evolving Skill-Graph. We first initialize the graph by refining human expert trajectories into atomic, natural-language skills. For execution, GeoSkill employs an inference model to perform direct reasoning guided by the current Skill-Graph. For continuous growth, an Autonomous Evolution mechanism leverages a larger model to conduct multiple reasoning rollouts on image-coordinate pairs sourced from web-scale data and verified real-world reasoning. By analyzing both successful and failed trajectories from these rollouts, the mechanism iteratively synthesizes and prunes skills, effectively expanding the Skill-Graph and correcting geographic biases without any parameter updates. Experiments demonstrate that GeoSkill achieves promising performance in both geolocation accuracy and reasoning faithfulness on GeoRC, while maintaining superior generalization across diverse external datasets. Furthermore, our autonomous evolution fosters the emergence of novel, verifiable skills, significantly enhancing the system's cognition of real-world geographic knowledge beyond isolated case studies.