As large language models (LLMs) are adapted to sensitive domains such as medicine, their fluency raises safety risks, particularly regarding provenance and accountability. Watermarking embeds detectable patterns to mitigate these risks, yet its reliability in medical contexts remains untested. Existing benchmarks focus on detection-quality tradeoffs and overlook factual risks. In medical text, watermarking often reweights low-entropy tokens, which are highly predictable and often carry critical medical terminology. Shifting these tokens can cause inaccuracy and hallucinations, risks that prior general-domain benchmarks fail to capture. We propose a medical-focused evaluation workflow that jointly assesses factual accuracy and coherence. Using GPT-Judger and further human validation, we introduce the Factuality-Weighted Score (FWS), a composite metric prioritizing factual accuracy beyond coherence to guide watermarking deployment in medical domains. Our evaluation shows current watermarking methods substantially compromise medical factuality, with entropy shifts degrading medical entity representation. These findings underscore the need for domain-aware watermarking approaches that preserve the integrity of medical content.