Large language models have achieved strong performance on medical reasoning benchmarks, yet their deployment in clinical settings demands rigorous verification to ensure factual accuracy. While reward models offer a scalable approach for reasoning trace verification, existing methods face two limitations: they produce only scalar reward values without explicit justification, and they rely on single-pass retrieval that precludes adaptive knowledge access as verification unfolds. We introduce Med-TIV, an agentic framework that addresses these limitations by training medical reward models to iteratively query external medical evidence during evaluation. Our approach combines tool-augmented verification with an iterative reinforcement learning paradigm that requires only trace-level supervision, alongside an adaptive curriculum mechanism that dynamically adjusts training data distribution. Across four medical reasoning benchmarks, Med-TIV achieves substantial gains over existing methods, improving MedQA accuracy by 23.5% and MedXpertQA by 32.0% relative to the base generator. Crucially, Med-TIV demonstrates an 8× reduction in sampling budget compared to prior reward model baselines. These findings establish that agentic medical reward modeling offers a principled path toward more reliable medical reasoning systems.