More frequent and intense mass bleaching events due to climate change have resulted in widespread coral mortality, which has selected for more thermally-tolerant corals. While adaptive alleles that promote thermal tolerance may be present in new coral recruits, the strong size-dependent demography of corals may slow population adaptation to climate change. For example, smaller, more thermally-tolerant coral recruits have lower survival, take years to become mature, only to produce relatively few offspring compared to adults. Therefore, understanding coral population evolution will require models that account for both size-related demography and genetic evolution. Here, we present the first size-structured evolutionarily-explicit projection model of reef-building corals to estimate coral adaptation to future projected climate change. The model is calibrated using past heat stress, and simulates future outcomes using heat stress profiles from four possible future pathways. Our model shows that with sufficient fecundity and recruitment during heat stress events, evolutionary rescue is possible. While only intermediate heritability in thermal tolerance will allow sufficient rates of evolution, eroding genetic variation may significantly reduce the rate of population adaptation and thus persistence to future climate change. This model will aid policymakers in determining rates of coral adaptation given different future pathways and interventions.