We evaluate the potential for current and future cosmic shear measurements from large galaxy surveys to constrain the impact of baryonic physics on the matter power spectrum. We do so using a model-independent parameterization that describes deviations of the matter power spectrum from the dark-matter-only case as a set of principal components that are localized in wavenumber and redshift. We perform forecasts for a variety of current and future datasets, and find that at least ~90% of the constraining power of these datasets is contained in no more than nine principal components. The constraining power of different surveys can be quantified using a figure of merit defined relative to currently available surveys. With this metric, we find that the final Dark Energy Survey dataset (DES Y5) and the Hyper Suprime Cam Survey will be roughly an order of magnitude more powerful than existing data in constraining baryonic effects. Upcoming Stage IV surveys (LSST, Euclid, and WFIRST) will improve upon this by a further factor of a few. We show that this conclusion is robust to marginalization over several key systematics. The ultimate power of cosmic shear to constrain galaxy formation is dependent on understanding systematics in the shear measurements at small (sub-arcminute) scales. If these systematics can be sufficiently controlled, cosmic shear measurements from DES Y5 and other future surveys have the potential to provide a very clean probe of galaxy formation and to strongly constrain a wide range of predictions from modern hydrodynamical simulations.

15 pages, 12 figures. v2: MNRAS published version, with references and clarifications added