Automatic medical image segmentation plays a crucial role in clinical diagnosis and treatment. However, it is still a challenging task due to the complex interior characteristics (e.g., inconsistent intensity, low contrast, texture heterogeneity) and ambiguous external boundary structures. In this paper, we introduce a novel geometric structure learning mechanism (GSLM) to overcome the limitations of existing segmentation models that lack learning ”focus, path, and difficulty.” The geometric structure in this mechanism is jointly characterized by the skeleton-like structure extracted by the mask distance transform (MDT) and the boundary structure extracted by the mask distance inverse transform (MDIT). Among them, the skeleton-like and boundary pay attention to the trend of interior characteristics consistency and external structure continuity, respectively. With this idea, we design GSAL, a novel end-to-end geometric structure adversarial learning for robust medical image segmentation. GSAL has four components: a geometric structure generator, which yields the geometric structure to learn the most discriminative features that preserve interior characteristics consistency and external boundary structure continuity, skeleton-like and boundary structure discriminators, which enhance and correct the characterization of internal and external geometry to mutually promote the capture of global contextual dependencies, and a geometric structure fusion sub-network, which fuses the two complementary and refined skeleton-like and boundary structures to generate the high-quality segmentation results. The proposed approach has been successfully applied to three different challenging medical image segmentation tasks, including polyp segmentation, COVID-19 lung infection segmentation, and lung nodule segmentation. Extensive experimental results demonstrate that the proposed GSAL achieves favorably against most state-of-the-art methods under different evaluation metrics. The code is available at: https://github.com/DLWK/GSAL.