We search for excess gamma-ray emission coincident with the positions of confirmed and candidate Milky Way satellite galaxies using six years of data from the Fermi Large Area Telescope (LAT). Our sample of 45 stellar systems includes 28 kinematically confirmed dark-matter-dominated dwarf spheroidal galaxies (dSphs) and 17 recently discovered systems that have photometric characteristics consistent with the population of known dSphs. For each of these targets, the relative predicted gamma-ray flux due to dark matter annihilation is taken from kinematic analysis if available, and estimated from a distance-based scaling relation otherwise, assuming that the stellar systems are DM-dominated dSphs. LAT data coincident with four of the newly discovered targets show a slight preference (each similar to 2 sigma local) for gamma-ray emission in excess of the background. However, the ensemble of derived gamma-ray flux upper limits for individual targets is consistent with the expectation from analyzing random blank-sky regions, and a combined analysis of the population of stellar systems yields no globally significant excess (global significance < 1 sigma). Our analysis has increased sensitivity compared to the analysis of 15 confirmed dSphs by Ackermann et al. The observed constraints on the DM annihilation cross section are statistically consistent with the background expectation, improving by a factor of similar to 2 for large DM masses (m(DM, b<(b)over bar>) greater than or similar to 1 TeV and m(DM, tau+tau-) greater than or similar to 70 GeV) and weakening by a factor of similar to 1.5 at lower masses relative to previously observed limits.