Hallucination detection in clinical large vision-language models requires cross-verifying heterogeneous evidence, including patient text and images for faithfulness, and medical knowledge for factuality. However, existing pipelines either verify these dimensions in isolation or conflate them into a uniform process, potentially causing evidence misalignment that limits detection precision. Consequently, we posit that the essence of black-box hallucination detection is an evidence routing problem and propose MedF²Route, an Evidence-Routed Graph Verification Framework for both Faithfulness and Factuality hallucination detection. Specifically, it first transforms free-text clinical queries and model responses into structured graphs, partitioning the response graph into independent subgraphs to separate patient-specific grounding from knowledge-based diagnostic inference. An evidence-aware routing and verification mechanism then dynamically dispatches response triplets in these subgraphs to their dedicated heterogeneous evidence sources for targeted verification. Experiments on our MedF²Bench diagnostic benchmark show that our framework outperforms state-of-the-art baselines, notably improving detection precision. Overall, MedF²Route enables professional and precise detection of clinical hallucinations, underpinning the trustworthiness of clinical LVLMs. The code and data are available at https://github.com/Agentic-CliniAI/MedF2Route.