The underwater acoustic sensor network (UASN) is crucial for monitoring marine environments and detecting underwater targets. Reliable networking, especially routing technique, is essential in the UASN due to the challenges of underwater acoustics. In this article, an intelligent routing method based on graph neural network (GNN), named as the GNN-IR method, is proposed to address complex network topology and adapt to dynamic network environment. The proposed GNN-IR method leverages the GNN's ability to process non-Euclidean data and its strong predictive capability. It integrates the UASN's characteristics by building a model of network state prediction and designing the corresponding functions. Additionally, a prediction validity period mechanism in the GNN-IR method is designed to balance packet delivery ratio (PDR) and network energy efficiency. Simulations show that the GNN-IR method performs well in terms of effective data transmission, long network lifetime, and low additional routing information overhead. Furthermore, the proposed routing method is demonstrated an enhanced adaptability and flexibility in the complex underwater environment.