Differential privacy (DP) provides formal protection for sensitive data but typically incurs substantial losses in diagnostic performance. Model initialization has emerged as a critical factor in mitigating this degradation, yet the role of modern self-supervised learning under full-model DP remains poorly understood. Here, we present a large-scale evaluation of initialization strategies for differentially private medical image analysis, using chest radiograph classification as a representative benchmark with more than 800,000 images. Using state-of-the-art ConvNeXt models trained with DP-SGD across realistic privacy regimes, we compare non-domain-specific supervised ImageNet initialization, non-domain-specific self-supervised DINOv3 initialization, and domain-specific supervised pretraining on MIMIC-CXR, the largest publicly available chest radiograph dataset. Evaluations are conducted across five external datasets spanning diverse institutions and acquisition settings. We show that DINOv3 initialization consistently improves diagnostic utility relative to ImageNet initialization under DP, but remains inferior to domain-specific supervised pretraining, which achieves performance closest to non-private baselines. We further demonstrate that initialization choice strongly influences demographic fairness, cross-dataset generalization, and robustness to data scale and model capacity under privacy constraints. The results establish initialization strategy as a central determinant of utility, fairness, and generalization in differentially private medical imaging.