Recent advances in Large Language Models (LLMs) have significantly transformed conversational AI, enabling their integration into sensitive domains such as healthcare. The growing integration of LLMs with search engines has accelerated this shift by replacing traditional symptom lookups with conversational queries. A notable development is the increasing trend in medical self-diagnosis by the general public, which raises critical concerns regarding accuracy, fairness, and responsible deployment. Unlike professional diagnoses conducted by clinical experts, self-diagnosis often depends on incomplete or subjective symptom descriptions, further complicated by unrestricted access to LLM-generated outputs. This study evaluates the feasibility and risks associated with public LLMs for self-diagnosis, focusing on prompt engineering strategies, demographic bias, and the application of Retrieval Augmented Generation (RAG) to enhance reliability. Through a comparative analysis of 10,000 synthetic patient cases across LLMs, we discuss significant inconsistencies and demographic disparities. Our findings underscore the limitations of current public LLMs for unsupervised medical use and demonstrate how RAG can improve diagnostic accuracy while mitigating bias-driven errors. To the best of our knowledge, this is the first study to systematically explore extrinsic bias and responsible AI considerations in the context of LLM-powered self-diagnosis, highlighting the urgent need for safeguards and deployment frameworks in consumer-facing health AI tools.