Existing end-to-end health rumor detection methods lack decision transparency and interpretability. While recent efforts incorporate external evidence, they typically integrate evidence through simple concatenation, failing to adequately capture the fine-grained semantic relationships between evidence and rumors. To address these issues, we propose an Evidence-Augmented Generative Explanation (EAGE) framework that explicitly models evidence-rumor relationships through a retrieve-generate-verify pipeline. First, we design a dual-channel retrieval engine combining DPR for semantic matching and BM25 for term matching. Reciprocal rank fusion is applied to re-rank the combined results, enhancing the relevance of retrieved scientific evidence. Next, we leverage a cloud-based LLM to generate label-guided explanations. The resulting (rumor, evidence, explanation) triplets form an instruction set for parameter-efficient fine-tuning of the local Llama-3-8B. Finally, a cross-attention mechanism quantifies the semantic alignment between generated explanation and the original rumor, producing a detection result and supporting rationale. Comparative experiments on two COVID-19 datasets demonstrate that the proposed method outperforms baselines in both accuracy and explainability.