Accurate and timely fault diagnosis is of great significance for the safe operation and power supply reliability of distribution systems. However, traditional intelligent methods limit the use of the physical structures and data information of power networks. To this end, this study proposes a fault diagnostic model for distribution systems based on deep graph learning. This model considers the physical structure of the power network as a significant constraint during model training, which endows the model with stronger information perception to resist abnormal data input and unknown application conditions. In addition, a special spatiotemporal convolutional block is utilized to enhance the waveform feature extraction ability. This enables the proposed fault diagnostic model to be more effective in dealing with both fault waveform changes and the spatial effects of faults. In addition, a multi-task learning framework is constructed for fault location and fault type analysis, which improves the performance and generalization ability of the model. The IEEE 33-bus and IEEE 37-bus test systems are modeled to verify the effectiveness of the proposed fault diagnostic model. Finally, different fault conditions, topological changes, and interference factors are considered to evaluate the anti-interference and generalization performance of the proposed model. Experimental results demonstrate that the proposed model outperforms other state-of-the-art methods.