Galaxy color gradients (CGs)—I.e., spectral energy distributions that vary across the galaxy profile—will impact galaxy shape measurements when the modeled point-spread function (PSF) corresponds to that for a galaxy with spatially uniform color. This paper describes the techniques and results of a study of the expected impact of galaxy CGs on weak lensing measurements with the Large Synoptic Survey Telescope (LSST) when the PSF size depends on wavelength. The bias on cosmic shear measurements from CGs is computed both for parametric bulge+disk galaxy simulations and for more realistic chromatic galaxy surface brightness profiles based on Hubble Space Telescope V- and I-band images in the All-Wavelength Extended Groth Strip International Survey (AEGIS). For the parametric galaxies, and for the more realistic galaxies derived from AEGIS galaxies with a sufficient signal-to-noise ratio that CG bias can be isolated, the predicted multiplicative shear biases due to CGs are found to be at least a factor of two below the LSST full-depth requirement on the total systematic uncertainty in the redshift-dependent shear calibration. The analysis code and data products are publicly available (https://github.com/sowmyakth/measure_cg_bias).