Large Language Models (LLMs) show promise for medical education, but most existing systems focus on localized interactions such as question answering or single-turn feedback, rather than organizing an entire clinical case into a decision-centered learning trajectory. We introduce \textit{MedGame}, a framework that transforms static clinical cases into structured, executable storytelling games. MedGame uses a dual-engine design: a Medical Narrative Designer synthesizes case-grounded clinical storylines with states and decision nodes, while a Story Director converts them into dependency-aware multimodal orchestration plans rendered by our released interactive platform. We construct MedGame Bench, a 5,000-case benchmark and evaluation protocol for Medical Narrative Generation and Story Direction. Experiments show that task-specific fine-tuning substantially improves open-source LLMs on MedGame Bench and narrows the gap with commercial models. A pilot student study further shows that learners perceive MedGame as more engaging and useful than text-only alternatives.