Despite recent Multimodal Large Language Models (MLLMs)' linguistic prowess in medical diagnosis, we find even state-of-the-art MLLMs suffer from a critical perceptual deficit: geometric blindness. This failure to ground outputs in objective geometric constraints leads to plausible yet factually incorrect hallucinations, rooted in training paradigms that prioritize linguistic fluency over geometric fidelity. This paper introduces Med-Scout, a novel framework that "cures" this blindness via Reinforcement Learning (RL) that leverages the intrinsic geometric logic latent within unlabeled medical images. Instead of relying on costly expert annotations, Med-Scout derives verifiable supervision signals through three strategic proxy tasks inspired by the systematic reading and reasoning patterns of clinicians: Hierarchical Scale Localization, Topological Jigsaw Reconstruction, and Anomaly Consistency Detection. To rigorously quantify this deficit, we present Med-Scout-Bench, a new benchmark specifically designed to evaluate geometric perception. Extensive evaluations show that Med-Scout significantly mitigates geometric blindness, outperforming leading proprietary and open-source MLLMs by over 40% on our benchmark. Furthermore, this enhanced geometric perception generalizes to broader medical understanding, achieving superior results on radiological and comprehensive medical VQA tasks.