Two facets of climate change--increased tropical storm intensity and ocean acidification--are expected to detrimentally affect reef-building organisms by increasing their mortality rates and decreasing their calcification rates. Our current understanding of these effects is largely based on individual organisms' short-term responses to experimental manipulations. However, predicting the ecologically-relevant effects of climate change requires understanding the long-term demographic implications of these organism-level responses. In this study, we investigate how storm intensity and calcification rate interact to affect population dynamics of the table coral Acropora hyacinthus, a dominant and geographically widespread ecosystem engineer on wave-exposed Indo-Pacific reefs. We develop a mechanistic framework based on the responses of individual-level demographic rates to changes in the physical and chemical environment, using a size-structured population model that enables us to rigorously incorporate uncertainty. We find that table coral populations are vulnerable to future collapse, placing in jeopardy many other reef organisms that are dependent upon them for shelter and food. Resistance to collapse is largely insensitive to predicted changes in storm intensity, but is highly dependent on the extent to which calcification influences both the mechanical properties of reef substrate and the colony-level trade-off between growth rate and skeletal strength. This study provides the first rigorous quantitative accounting of the demographic implications of the effects of ocean acidification and changes in storm intensity, and provides a template for further studies of climate-induced shifts in ecosystems, including coral reefs.