Self-supervised learning (SSL) has transformed vision encoder training in general domains but remains underutilized in medical imaging due to limited data and domain specific biases. We present MammoDINO, a novel SSL framework for mammography, pretrained on 1.4 million mammographic images. To capture clinically meaningful features, we introduce a breast tissue aware data augmentation sampler for both image-level and patch-level supervision and a cross-slice contrastive learning objective that leverages 3D digital breast tomosynthesis (DBT) structure into 2D pretraining. MammoDINO achieves state-of-the-art performance on multiple breast cancer screening tasks and generalizes well across five benchmark datasets. It offers a scalable, annotation-free foundation for multipurpose computer-aided diagnosis (CAD) tools for mammogram, helping reduce radiologists' workload and improve diagnostic efficiency in breast cancer screening.