Recent studies have shown that unsupervised graph contrastive learning (GCL) is vulnerable to adversarial attacks. Automatic adversarial augmentation techniques are proposed to improve both the effectiveness and robustness of GCL. Existing methods typically regard unsupervised contrastive loss as the adversarial goal, essentially aiming to maximize inter-view instance-wise discrepancies between adversarial and original views. However, such attacks overlook intra-view neighborhood inconsistency, which hinders the robustness of GCL models against local neighborhood noises, resulting in performance degradation on low-homophily graphs. To tackle this issue, we propose a novel adversarial contrastive paradigm, named Edge self-aDversarial Augmentation for Graph Contrastive Learning (EDA-GCL). We theoretically establish that the adversarial objective of the intra-view neighborhood is equivalent to maximizing the discrepancy between bidirectional edge features. Hence, we build our adversarial framework based on edge self-adversarial learning. It generates pairwise adversarial augmentations from the original view by learning distinct neighborhood connectivity structures. The learned pairwise adversarial views are utilized for GCL model training in the minimization stage. Notably, this edge-level adversarial approach reduces the computational complexity to the level of the edge number. Experiments on various graph tasks and complex noise scenarios demonstrate the superiority and robustness of our EDA-GCL.