Generative AI has emerged as a transformative force in health care, driving advancements in medical diagnostics, personalized treatments, and operational efficiencies. As this technology becomes increasingly prevalent, it brings both opportunities and challenges, particularly concerning fairness and bias. This position paper delves into the impact of generative AI on health care by first unveiling the hidden challenges of bias and its implications for equitable medical outcomes. We present a balanced debate on the advantages and disadvantages of bias mitigation strategies, providing a critical analysis of their effectiveness and feasibility. Our exploration includes current recommendations aimed at ensuring fairness in AI applications. Through real-world examples, we illustrate how bias has been successfully addressed and the lessons learned. By integrating these elements, this position paper offers a comprehensive perspective on how generative AI can be harnessed to improve health care while navigating the complexities of fairness and equity.