Recent analyses have shown that the concluding stages of giant planet formation are accompanied by the development of large-scale meridional flow of gas inside the planetary Hill sphere. This circulation feeds a circumplanetary disk that viscously expels gaseous material back into the parent nebula, maintaining the system in a quasi-steady state. Here we investigate the formation of natural satellites of Jupiter and Saturn within the framework of this newly outlined picture. We begin by considering the long-term evolution of solid material, and demonstrate that the circumplanetary disk can act as a global dust trap, where s ~ 0.1-10 mm grains achieve a hydrodynamical equilibrium, facilitated by a balance between radial updraft and aerodynamic drag. This process leads to a gradual increase in the system’s metallicity, and eventually culminates in the gravitational fragmentation of the outer regions of the solid sub-disk into R ~ 100 km satellitesimals. Subsequently, satellite conglomeration ensues via pairwise collisions, but is terminated when disk-driven orbital migration removes the growing objects from the satellitesimal feeding zone. The resulting satellite formation cycle can repeat multiple times, until it is brought to an end by photo-evaporation of the parent nebula. Numerical simulations of the envisioned formation scenario yield satisfactory agreement between our model and the known properties of the Jovian and Saturnian moons.