Vision-based surgical skill assessment (SSA) enables objective and scalable evaluation of operative performance. Progress in this field is constrained by the high cost and time demands for manual annotation of quantitative skill scores, as well as the poor generalization of existing regression models to new surgical tasks and environments. Meanwhile, appreciable volumes of unlabeled video data are now available, motivating the development of unsupervised domain adaptation (UDA) methods for SSA. We introduce the first benchmark for UDA in SSA regression, spanning four datasets across dry-lab and clinical settings as well as open and robotic surgery. We evaluate eight representative models under challenging domain shifts and propose CoRe-DA, a novel contrastive regression-based adaptation framework. Our method learns domain-invariant representations through relative-score supervision and target-domain self-training. Comprehensive experiments across two UDA settings show that CoRe-DA is superior to state-of-the-art methods, achieving Spearman Correlation Coefficients of 0.46 and 0.41 on dry-lab and clinical target datasets, respectively, without using any labeled target data for training. Overall, CoRe-DA enables scalable SSA with reliable cross-domain generalization, where existing methods underperform. Our code and datasets will be released at https://github.com/anastadimi/CoRe-DA.