Fact verification (FV) aims to assess the veracity of a claim based on relevant evidence. The traditional approach for automated FV includes a three-part pipeline relying on short evidence snippets and encoder-only inference models. More recent approaches leverage the multi-turn nature of LLMs to address FV as a step-by-step problem where questions inquiring additional context are generated and answered until there is enough information to make a decision. This iterative method makes the verification process rational and explainable. While these methods have been tested for encyclopedic claims, exploration on domain-specific and realistic claims is missing. In this work, we apply an iterative FV system on three medical fact-checking datasets and evaluate it with multiple settings, including different LLMs, external web search, and structured reasoning using logic predicates. We demonstrate improvements in the final performance over traditional approaches and the high potential of step-by-step FV systems for domain-specific claims.