Healthcare artificial intelligence systems often degrade in performance when deployed across institutions, with documented performance drops and perpetuation of discriminatory patterns embedded in data. This brittleness comes, in part, from learning statistical associations rather than causal mechanisms. Causal graph neural networks address this by combining graph-based representations of biomedical data with causal inference to learn invariant mechanisms instead of just spurious correlations. This Perspective reviews the methodology of structural causal models, disentangled causal representation learning, and techniques for interventional prediction and counterfactual reasoning on graphs. We discuss applications across psychiatric diagnosis and brain network analysis, cancer subtyping with multi-omics causal integration, continuous physiological monitoring, and drug recommendations. These methods provide building blocks for patient-specific Causal Digital Twins that could support in silico clinical experimentation. Remaining challenges include computational costs that preclude real-time deployment, validation challenges that go beyond standard cross-validation, and the risk of causal-washing where methods adopt causal terminology without rigorous evidentiary support. We propose a tiered framework distinguishing causally-inspired architectures from causally-validated discoveries and outline future directions, including scalable causal discovery, multi-modal data integration, and regulatory pathways for these methods. Making practical Causal Digital Twins possible will require an honest assessment of what current methods deliver, sustained collaboration across disciplines, and validation standards that match the strength of the causal claims being made.