Most graph contrastive learning (GCL) methods heavily rely on cross-view contrast, thus facing several concomitant challenges, such as the complexity of designing effective augmentations, the potential for information loss between views, and increased computational costs. To mitigate reliance on cross-view contrasts, we propose \ttt{SIGNA}, a novel single-view graph contrastive learning framework. Regarding the inconsistency between structural connection and semantic similarity of neighborhoods, we resort to soft neighborhood awareness for GCL. Specifically, we leverage dropout to obtain structurally-related yet randomly-noised embedding pairs for neighbors, which serve as potential positive samples. At each epoch, the role of partial neighbors is switched from positive to negative, leading to probabilistic neighborhood contrastive learning effect. Furthermore, we propose a normalized Jensen-Shannon divergence estimator for a better effect of contrastive learning. Surprisingly, experiments on diverse node-level tasks demonstrate that our simple single-view GCL framework consistently outperforms existing methods by margins of up to 21.74% (PPI). In particular, with soft neighborhood awareness, SIGNA can adopt MLPs instead of complicated GCNs as the encoder to generate representations in transductive learning tasks, thus speeding up its inference process by 109 times to 331 times. The source code is available at https://github.com/sunisfighting/SIGNA.