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 models 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 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.