Large visual language models (VLMs) have shown strong multi-modal medical reasoning ability, but most operate as end-to-end black boxes, diverging from clinicians' evidence-based, staged workflows and hindering clinical accountability. Complementarily, expert visual grounding models can accurately localize regions of interest (ROIs), providing explicit, reliable evidence that improves both reasoning accuracy and trust. In this paper, we introduce CARE, advancing Clinical Accountability in multi-modal medical Reasoning with an Evidence-grounded agentic framework. Unlike existing approaches that couple grounding and reasoning within a single generalist model, CARE decomposes the task into coordinated sub-modules to reduce shortcut learning and hallucination: a compact VLM proposes relevant medical entities; an expert entity-referring segmentation model produces pixel-level ROI evidence; and a grounded VLM reasons over the full image augmented by ROI hints. The VLMs are optimized with reinforcement learning with verifiable rewards to align answers with supporting evidence. Furthermore, a VLM coordinator plans tool invocation and reviews evidence-answer consistency, providing agentic control and final verification. Evaluated on standard medical VQA benchmarks, our CARE-Flow (coordinator-free) improves average accuracy by 10.9% over the same size (10B) state-of-the-art (SOTA). With dynamic planning and answer review, our CARE-Coord yields a further gain, outperforming the heavily pre-trained SOTA by 5.2%. Our experiments demonstrate that an agentic framework that emulates clinical workflows, incorporating decoupled specialized models and explicit evidence, yields more accurate and accountable medical AI. Project page: https://xypb.github.io/CARE-Project-Page/