The classification of synthetic aperture radar (SAR) image scenes is widely used in military and civilian fields. However, some SAR images pose challenges in feature extraction due to complex scenes and low image resolution, making accurate SAR image classification difficult. To address this, an SAR image scene classification method based on the multi-layer fuzzy convolutional gray level co-occurrence matrix network (ML-FGLCMNet) approach is proposed. The multi-layer convolutional network module in this method can extract low-resolution features from SAR images, and the fuzzy gray-level co-occurrence matrix module combined four sets of second-order statistics can collect texture features from multiple angles. These features, along with the multi-class support vector machine classifier, are used to achieve SAR image scene classification. The Terra SAR-X and GS-SAR6 datasets were used for experimental analysis and comparison with the full-band channel attention network, multi-feature fusion global-local convolutional network and our method of extracting features with only the multi-layer convolutional network. The experimental results indicate that our method, ML-FGLCMNet, has good accuracy and Kappa index, which can reach more than 90% and 80%, respectively. In the confusion matrix, the classification accuracy of various SAR remote sensing scenes is higher than 0.86. It achieves smaller initial loss, higher training accuracy, and faster convergence compared to the comparative approaches.