Learning a self-supervised Monocular Depth Estimation (MDE) model with great generalization remains significantly challenging. Despite the success of adversarial augmentation in the supervised learning generalization, naively incorporating it into self-supervised MDE methods potentially causes over-regularization, suffering from severe performance degradation. In this paper, we conduct qualitative analysis and illuminate the main causes: (i) inherent sensitivity in the UNet-alike depth network and (ii) dual optimization conflict caused by over-regularization. To tackle these issues, we propose a general adversarial training framework, named Stabilized Conflict-optimization Adversarial Training (SCAT), integrating adversarial data augmentation into self-supervised MDE methods to achieve a balance between stability and generalization. Specifically, we devise an effective Scaling Depth Network that tunes the coefficients of long skip connection in DepthNet and effectively stabilizes the training process. Then, we propose a Conflict Gradient Surgery strategy, which progressively integrates the adversarial gradient and optimizes the model toward a conflict-free direction. Extensive experiments on five benchmarks demonstrate that SCAT can achieve state-of-the-art performance and significantly improve the generalization capability of existing self-supervised MDE methods.