Among health care researchers, there is increasing debate over how best to assess and ensure the fairness of algorithms used for clinical decision support and population health, particularly concerning potential racial bias. Here we first distill concerns over the fairness of health care algorithms into four broad categories: (a) the explicit inclusion (or, conversely, the exclusion) of race and ethnicity in algorithms, (b) unequal algorithm decision rates across groups, (c) unequal error rates across groups, and (d) potential bias in the target variable used in prediction. With this taxonomy, we critically examine seven prominent and controversial health care algorithms. We show that popular approaches that aim to improve the fairness of health care algorithms can in fact worsen outcomes for individuals across all racial and ethnic groups. We conclude by offering an alternative, consequentialist framework for algorithm design that mitigates these harms by instead foregrounding outcomes and clarifying trade-offs in the pursuit of equitable decision-making.