Domain shift is a commonly encountered issue in medical imaging solutions, primarily caused by variations in imaging devices and data sources. To mitigate this problem, unsupervised domain adaptation techniques have been employed. However, concerns regarding patient privacy and potential degradation of image quality have led to an increased focus on source-free domain adaptation. In this study, we address the issue of false labels in self-training based source-free domain adaptive medical image segmentation methods. To correct erroneous pseudo-labels, we propose a novel approach called the local-global pseudo-label correction (LGDA) method for source-free domain adaptive medical image segmentation. Our method consists of two components: An offline local context-based pseudo-label correction method that utilizes local context similarity in image space. And an online global pseudo-label correction method based on class prototypes, which corrects erroneously predicted pseudo-labels by considering the relative distance between pixel-wise feature vectors and prototype vectors. We evaluate the performance of our method on three benchmark fundus image datasets for optic disc and cup segmentation. Our method achieves superior performance compared to the state-of-the-art approaches, even without using of any source data.