Medical image segmentation is a crucial task in computer-aided diagnosis. Although convolutional neural networks (CNNs) have made significant progress in the field of medical image segmentation, the convolution kernels of CNNs are optimized from random initialization without explicitly encoding gradient information, leading to a lack of specificity for certain features, such as blurred boundary features. Furthermore, the frequently applied down-sampling operation also loses the fine structural features in shallow layers. Therefore, we propose a boundary-aware gradient operator network (BG-Net) for medical image segmentation, in which the gradient convolution (GConv) and the boundary-aware mechanism (BAM) modules are developed to simulate image boundary features and the remote dependencies between channels. The GConv module transforms the gradient operator into a convolutional operation that can extract gradient features; it attempts to extract more features such as images boundaries and textures, thereby fully utilizing limited input to capture more features representing boundaries. In addition, the BAM can increase the amount of global contextual information while suppressing invalid information by focusing on feature dependencies and the weight ratios between channels. Thus, the boundary perception ability of BG-Net is improved. Finally, we use a multi-modal fusion mechanism to effectively fuse lightweight gradient convolution and U-shaped branch features into a multilevel feature, enabling global dependencies and low-level spatial details to be effectively captured in a shallower manner. We conduct extensive experiments on eight datasets that broadly cover medical images to evaluate the effectiveness of the proposed BG-Net. The experimental results demonstrate that BG-Net outperforms the state-of-the-art methods, particularly those focused on boundary segmentation.