Climate change threatens organisms in a variety of interactive ways that requires simultaneous adaptation of multiple traits. Predicting evolutionary responses requires an understanding of the potential for interactions among stressors and the genetic variance and covariance among fitness‐related traits that may reinforce or constrain an adaptive response. Here we investigate the capacity of Acropora millepora, a reef‐building coral, to adapt to multiple environmental stressors: rising sea surface temperature, ocean acidification, and increased prevalence of infectious diseases. We measured growth rates (weight gain), coral color (a proxy for Symbiodiniaceae density), and survival, in addition to nine physiological indicators of coral and algal health in 40 coral genets exposed to each of these three stressors singly and combined. Individual stressors resulted in predicted responses (e.g., corals developed lesions after bacterial challenge and bleached under thermal stress). However, corals did not suffer substantially more when all three stressors were combined. Nor were trade‐offs observed between tolerances to different stressors; instead, individuals performing well under one stressor also tended to perform well under every other stressor. An analysis of genetic correlations between traits revealed positive covariances, suggesting that selection to multiple stressors will reinforce rather than constrain the simultaneous evolution of traits related to holobiont health (e.g., weight gain and algal density). These findings support the potential for rapid coral adaptation under climate change and emphasize the importance of accounting for corals’ adaptive capacity when predicting the future of coral reefs.