Physics-Informed Machine Learning (PIML) has shown substantial promise in accelerating computational mechanics by integrating physical laws into data-driven models. However, high computational cost and poor scalability, coupled with the geometry-specific nature of these problems, make achieving parametric and geometric generalizability a persistent challenge. Conventional PIML approaches often depend on coordinate-based representations and pointwise enforcement of governing equations, limiting their ability to generalize across unseen geometries and mesh resolutions. Graph neural networks (GNNs), with their message-passing operations between neighboring nodes, are naturally well suited for CFD problems governed by local interactions. Recent studies have demonstrated GNN-based models’ accuracy in learning flow fields on irregular meshes and generalizing across variations in Reynolds number, boundary conditions, and geometric configurations [1], [2], [3]. Despite these advances, current methods implicitly rely on large, geometry-diverse training datasets, which are expensive to generate and hinder scalability to new designs. Thus, a crucial question remains: can GNNs trained on a single geometry generalize to a priori unseen geometries when augmented with suitable geometric descriptors and physics-based constraints? Addressing this would enable data-efficient learning frameworks without extensive multi-geometry training. This work presents a physics-assisted GNN framework that encodes computational mesh topology and geometry-aware latent representations to facilitate strong geometric generalizability. Physical consistency is imposed via integral conservation laws—specifically, global mass conservation across multiple cross-stream control planes—making the approach robust to flow separation, recirculation, and geometric discontinuities, as seen in backward-facing step (BFS) flows. Benchmarking on BFS, the method enforces inlet and wall boundary conditions through hard constraints, ensuring physically admissible solutions. Validation on steady incompressible BFS flows across various Reynolds numbers demonstrates accurate velocity field predictions, and strong extrapolation to a priori unseen step heights, despite training on a single geometry. The framework offers a scalable, data-efficient alternative to conventional PINN and operator-learning approaches.