(Abridged) The violent hierarchical nature of the LCDM cosmology poses serious difficulties for the formation of disk galaxies. To help resolve these issues, we describe a new, merger-driven scenario for the cosmological formation of disk galaxies at high redshifts that supplements the standard model based on dissipational collapse.In this picture, large gaseous disks may be produced from high-angular momentum mergers of systems that are gas-dominated, i.e. M_gas/(M_gas +M_star > 0.5 at the height of the merger. Pressurization from the multiphase structure of the interstellar medium prevents the complete conversion of gas into stars during the merger, and if enough gas remains to form a disk, the remnant eventually resembles a disk galaxy. We perform numerical simulations of galaxy mergers to study how supernovae feedback strength, supermassive black hole growth and feedback, progenitor gas fraction, merger mass-ratio, and orbital geometry impact the formation of remnant disks. We find that disks can build angular momentum through mergers and the degree of rotational support of the baryons in the merger remnant is primarily related to feedback processes associated with star formation. Disk-dominated remnants are restricted to form in mergers that are gas-dominated at the time of final coalescence and gas-dominated mergers typically require extreme progenitor gas fractions (>80%). We also show that the formation of rotationally-supported stellar systems in mergers is not restricted to idealized orbits, or major or minor mergers. We suggest that the hierarchical nature of the LCDM cosmology and the physics of the interstellar gas may act together to form spiral galaxies by building the angular momentum of disks through early, gas-dominated mergers.