Stroke is an acute cerebrovascular disease, and timely diagnosis significantly improves patient survival. However, existing automated diagnosis methods suffer from fairness issues across demographic groups, potentially exacerbating healthcare disparities. In this work, we address this critical challenge by proposing FAST-CAD, a theoretically-grounded framework that integrates Domain-Adversarial Training (DAT) with Group Distributionally Robust Optimization (Group-DRO) for fair and accurate non‑contact stroke diagnosis. Our approach is built upon rigorous theoretical foundations from domain adaptation theory and minimax fairness, providing convergence guarantees and fairness bounds. We establish a comprehensive multimodal dataset encompassing 12 demographic subgroups defined by age, gender, and posture combinations. FAST-CAD employs self-supervised encoders with adversarial domain discrimination to learn demographic-invariant representations, while Group-DRO ensures robust performance across all subgroups by optimizing worst‑group risk. Extensive experiments demonstrate that our method achieves superior diagnostic performance (91.2\% AUC) while maintaining fairness across all demographic groups, with theoretical validation confirming the effectiveness of our unified DAT+Group-DRO framework. Our work provides both practical advances and theoretical insights for fair medical AI systems.