Naval shipboard power systems (SPS) are rapidly embracing electrification, resulting in loads that generate pulsation currents and encounter substantial transients. However, conventional time-based features alone are inadequate for effectively monitoring and safeguarding these loads against faults. This highlights the critical requirement for advanced machine learning based methods to discern and differentiate between the various transient stages within the load profile. In this paper, we propose a Wavelet Graph Neural Network (WGNN) model for non-intrusive fault detection in SPS. The fault detection system leverages the dynamic model of the SPS to train and test performance with varying fault scenarios. The underlying structure and the interdependence among component states in the SPS network are effectively captured using the WGNN model, resulting in accuracies over 99% for intrusive fault detection and 97% for non-intrusive fault detection. The developed WGNN model has also shown to be robust in the presence of pulse loads and noise, achieving an accuracy of over 95%. At the end, a real-time simulation of the proposed method is validated on a hardware-in-the-loop system, guaranteeing the high fidelity and low latency of the proposed approach. These findings validate the effectiveness of the proposed WGNN model for fault detection and real-world applications in SPS.