We present measurements of the gas-phase abundance ratio C/O in six H II regions in the spiral galaxies M101 and NGC 2403, based on ultraviolet spectroscopy using the Faint Object Spectrograph on the Hubble Space Telescope. The ratios of C to O increase systematically with O/H in both galaxies, from log(C/O){~=}-0.8 at log(O/H){~=}-4.0 to log(C/O){~=}-0.1 at log(O/H){~=}-3.4. C/N shows no correlation with O/H. The rate of increase of C/O is somewhat uncertain because of uncertainty as to the appropriate UV reddening law and uncertainty in the metallicity dependence on grain depletions. However, the trend of increasing C/O with O/H is clear, confirming and extending the trend in C/O indicated previously from observations of irregular galaxies. Our data indicate that the radial gradients in C/H across spiral galaxies are steeper than the gradients in O/H. Comparing the data to chemical-evolution models for spiral galaxies shows that models in which the massive star yields do not vary with metallicity predict radial C/O gradients that are much flatter than the observed gradients.