We examine formation conditions for the Galilean satellites in the context of models of late-stage giant planet accretion and satellite-disk interactions. We first reevaluate the current standard, in which the satellites form from a ``minimum mass subnebula'' disk, obtained by augmenting the mass of the current satellites to solar abundance and resulting in a disk mass containing about 2% of Jupiter's mass. Conditions in such a massive and gas-rich disk are difficult to reconcile with both the icy compositions of Ganymede and Callisto and the protracted formation time needed to explain Callisto's apparent incomplete differentiation. In addition, we argue that disk torques in such a gas-rich disk would cause large satellites to be lost to inward decay onto the planet. These issues have prevented us from identifying a self-consistent scenario for the formation and survival of the Galilean satellites using the standard model. We then consider an alternative, in which the satellites form in a circumplanetary accretion disk produced during the very end stages of gas accretion onto Jupiter. In this case, an amount of gas and solids of at least ~0.02 Jovian masses must still be processed through the disk during the satellite formation era, but this amount need not have been present all at once. We find that an accretion disk produced by a slow inflow of gas and solids, e.g., 2×10-7 Jovian masses per year, is most consistent with conditions needed to form the Galilean satellites, including disk temperatures low enough for ices and protracted satellite accretion times of >=105 yr. Such a ``gas-starved'' disk has an orders-of-magnitude lower gas surface density than the minimum mass subnebula (and for many cases is optically thin). Solids delivered to the disk build up over many disk viscous cycles, resulting in a greatly reduced gas-to-solids ratio during the final stages of satellite accretion. This allows for the survival of Galilean-sized satellites against disk torques over a wide range of plausible conditions.