Coronary artery calcium (CAC) scoring is a key predictor of cardiovascular risk, but it relies on ECG-gated CT scans, restricting its use to specialized cardiac imaging settings. We introduce an automated framework for CAC detection and lesion-specific Agatston scoring that operates across both gated and non-gated CT scans. At its core is CARD-ViT, a self-supervised Vision Transformer trained exclusively on gated CT data using DINO. Without any non-gated training data, our framework achieves 0.707 accuracy and a Cohen's kappa of 0.528 on the Stanford non-gated dataset, matching models trained directly on non-gated scans. On gated test sets, the framework achieves 0.910 accuracy with Cohen's kappa scores of 0.871 and 0.874 across independent datasets, demonstrating robust risk stratification. These results demonstrate the feasibility of cross-domain CAC scoring from gated to non-gated domains, supporting scalable cardiovascular screening in routine chest imaging without additional scans or annotations.