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Among healthcare 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 healthcare algorithms into four broad categories: (1) the explicit inclusion (or, conversely, the exclusion) of race and ethnicity in algorithms, (2) unequal algorithm decision rates across groups, (3) unequal error rates across groups, and (4) potential bias in the target variable used in prediction. With this taxonomy, we critically examine seven prominent and controversial healthcare algorithms. We show that popular approaches that aim to improve the fairness of healthcare 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 tradeoffs in the pursuit of equitable decision making.