AI agents with tool-use capabilities show promise for integrating the domain expertise of various tools. In the medical field, however, tools are usually AI models that are inherently error-prone and can produce contradictory responses. Existing research on medical agents lacks sufficient understanding of the tools' realistic reliability and thus cannot effectively resolve tool conflicts. To address this gap, this paper introduces a framework that enables an agent to interact with tools and empirically learn their practical trustworthiness across different types of multimodal queries via agentic learning. As a concrete instantiation, we focus on chest X-ray analysis and present a tool-expertise-aware chest X-ray agent (TEA-CXA). When tool outputs disagree, the agent experimentally accepts or rejects multimodal tool results, receives rewards, and learns which tool to trust for each query type. Importantly, TEA-CXA extends existing codebases for reinforcement learning with multi-turn tool-calling that focus on textual inputs, to support multimodal contexts effectively. In addition, we enhance the codebase for medical use scenarios by supporting multiple tool calls in one turn, parallel tool inference, and multi-image accommodation within a single user query. Our code framework is applicable to general medical research on multi-turn tool-calling reinforcement learning in multimodal settings. Experiments show that TEA-CXA outperforms the state-of-the-art methods and a comprehensive set of baselines. Code will be released.