We have carried out an H i stacking analysis of a volume-limited sample of ∼5000 galaxies with imaging and spectroscopic data from GALEX and the Sloan Digital Sky Survey, which lie within the current footprint of the Arecibo Legacy Fast ALFA (ALFALFA) survey. Our galaxies are selected to have stellar masses greater than 1010 M⊙ and redshifts in the range 0.025 < z < 0.05. We extract a subsample of 1833 ‘early-type’ galaxies with inclinations less than 70°, with concentration indices C > 2.6 and with light profiles that are well fit by a De Vaucouleurs model. We then stack H i line spectra extracted from the ALFALFA data cubes at the 3D positions of the galaxies from these two samples in bins of stellar mass, stellar mass surface density, central velocity dispersion and NUV−r colour. We use the stacked spectra to estimate the average H i gas fractions /M* of the galaxies in each bin. Our main result is that the H i content of a galaxy is not influenced by its bulge. The average H i gas fractions of galaxies in both our samples correlate most strongly with NUV−r colour and with stellar surface density. The relation between average H i fraction and these two parameters is independent of concentration index C. We have tested whether the average H i gas content of bulge-dominated galaxies on the red sequence differs from that of late-type galaxies on the red sequence. We find no evidence that galaxies with a significant bulge component are less efficient at turning their available gas reservoirs into stars. This result is in contradiction with the ‘morphological quenching’ scenario proposed by Martig et al.