The performance of predictive models in clinical settings often degrades when deployed in new hospitals due to distribution shifts. This paper presents a large-scale study of causality-inspired domain generalization on heterogeneous multi-center intensive care unit (ICU) data. We apply anchor regression and introduce anchor boosting, a novel, tree-based nonlinear extension, to a large dataset comprising 400,000 patients from nine distinct ICU databases. We find that anchor regularization yields improvements of out-of-distribution performance, particularly for the most dissimilar target domains. The methods appear robust to violations of theoretical assumptions, such as anchor exogeneity. Furthermore, we propose a novel conceptual framework to quantify the utility of large external data datasets. By evaluating performance as a function of available target-domain data, we identify three regimes: (i) a domain generalization regime, where only the external model should be used, (ii) a domain adaptation regime, where refitting the external model is optimal, and (iii) a data-rich regime, where external data provides no additional value.