Future changes in the North American monsoon, a circulation system that brings abundant summer rains to vast areas of the North American Southwest, could have significant consequences for regional water resources. How this monsoon will change with increasing greenhouse gases, however, remains unclear, not least because coarse horizontal resolution and systematic sea-surface temperature biases limit the reliability of its numerical model simulations. Here we investigate the monsoon response to increased atmospheric carbon dioxide (CO$_2$) concentrations using a 50-km-resolution global climate model which features a realistic representation of the monsoon and its synoptic-scale variability. It is found that the monsoon response to CO$_2$ doubling is sensitive to sea-surface temperature biases. When minimizing these biases, the model projects a robust reduction in monsoonal precipitation over the southwestern United States, contrasting with previous multi-model assessments. Most of this precipitation decline can be attributed to increased atmospheric stability, and hence weakened convection, caused by uniform sea-surface warming. These results suggest improved adaptation measures, particularly water resource planning, will be required to cope with projected reductions in monsoon rainfall in the American Southwest.