Equitable access to reliable health information is vital for public health, but the quality of online health resources varies by language, raising concerns about inconsistencies in Large Language Models (LLMs) for healthcare. In this study, we examine the consistency of responses provided by LLMs to health-related questions across English, German, Turkish, and Chinese. We largely expand the HealthFC dataset by categorizing health-related questions by disease type and broadening its multilingual scope with Turkish and Chinese translations. We reveal significant inconsistencies in responses that could spread healthcare misinformation. Our main contributions are 1) a multilingual health-related inquiry dataset with meta-information on disease categories, and 2) a novel prompt-based evaluation workflow that enables sub-dimensional comparisons between two languages through parsing. Our findings highlight key challenges in deploying LLM-based tools in multilingual contexts and emphasize the need for improved cross-lingual alignment to ensure accurate and equitable healthcare information.