Modern gaze estimation models can accurately predict human gaze from facial images. However, due to privacy concerns and intricate data collection procedures, gaze estimation datasets are typically smaller and less diverse compared to those for other vision tasks, which directly leads to poor generalization in gaze estimation models. Common solutions, such as domain adaptation models, require additional domain-specific data, yet such data is often difficult to obtain due to privacy restrictions. Meanwhile, domain generalization models suffer from limited performance due to insufficient training data. To address these fundamental challenges---privacy and data diversity---we explore privacy-preserving gaze data generation schemes and propose a novel data-driven generalization solution. Specifically, we develop two diffusion-based generative models, DDPM-Gaze and LDM-Gaze, for synthesizing gaze data. We demonstrate that synthetic data can significantly improve generalization performance when simply used with fine-tuning-based methods. Furthermore, we introduce the Domain Stability Adaptation (DSA) framework, a simple yet effective domain generalization approach that enhances model robustness by increasing the domain uncertainty of input samples while reducing prediction uncertainty. Extensive experiments validate the effectiveness of our synthetic data and demonstrate the superiority of our data-driven generalization solution.