In recent years, Large Language Models (LLMs) have become widely used in medical applications, such as clinical decision support, medical education, and medical question answering. Yet, these models are often English-centric, limiting their robustness and reliability for linguistically diverse communities. Recent work has highlighted discrepancies in performance in low-resource languages for various medical tasks, but the underlying causes remain poorly understood. In this study, we conduct a cross-lingual empirical analysis of LLM performance on Arabic and English medical question and answering. Our findings reveal a persistent language-driven performance gap that intensifies with increasing task complexity. Tokenization analysis exposes structural fragmentation in Arabic medical text, while reliability analysis suggests that model-reported confidence and explanations exhibit limited correlation with correctness. Together, these findings underscore the need for language-aware design and evaluation strategies in LLMs for medical tasks.