River routing provides the critical link between runoff generation and downstream streamflow prediction, yet conventional routing models often rely on simplified hydraulic assumptions, fixed parameters, and conservative transport formulations that may limit performance in heterogeneous river systems. Here, we introduce <em>RiverGraphNet</em>, a physics-aware graph-based routing framework designed to route physically generated gridded runoff through directed river networks. The framework explicitly isolates routing from runoff generation by coupling <em>Noah-MP</em> runoff with a directed river-network graph derived from the <em>NextGen</em> hydrofabric for the <em>Salt–Verde</em> watershed (Arizona, USA). Gridded runoff is transferred from the Noah-MP domain to graph nodes representing hydrologic routing elements, while streamflow propagation is learned using a <em>Graph Attention Network</em> (GAT) informed by physically meaningful node and edge attributes describing drainage structure, terrain, and hydraulic-routing proxies.</p> <p><em>RiverGraphNet</em> was evaluated against observed daily streamflow at 23 USGS gauges and compared with RAPID, a widely used physics-based routing benchmark forced with identical <em>Noah-MP</em> gridded runoff inputs. Experiments examined the influence of temporal routing memory (3–30 day runoff lags), temporal convolution, and alternative loss functions (MSE, weighted MSE, and JKGE-based objectives). <em>RiverGraphNet</em> consistently outperformed RAPID across nearly all gauges and configurations. The best-performing experiment (14-day lag weighted MSE) achieved a median KG<em>E</em><sub>{<em>ss</em>}</sub> of 0.72, substantially exceeding RAPID performance (median KG<em>E</em><sub>{<em>ss</em>} </sub>of 0.0). Event-scale hydrographs and flow-duration analyses demonstrated improved representation of <em>peak timing, event magnitude</em>, and <em>long-term streamflow distributions</em>. Attention analysis further revealed that routing improvements were most strongly associated with channel-width and dominant-pathway metrics rather than travel-time proxies alone, suggesting that adaptive representation of network influence provides predictive value beyond conventional travel-time parameterization. These results demonstrate the potential of physics-aware, topology-constrained graph learning as a flexible alternative for routing gridded hydrologic runoff through complex river networks.