We examine X-ray emission produced from hot gas during collisions and mergers of disk galaxies. To study this process, we employ simulations that incorporate cosmologically motivated disk-galaxy models and include the effects of radiative cooling, star formation, supernova feedback, and accreting supermassive black holes. We find that during a merger, the colliding gas in the disks is shock-heated to X-ray-emitting temperatures. The X-ray luminosity is spatially extended, rises during the initial stages of the merger, and peaks when the galactic centers coalesce. When a physical model for accreting black holes is included, the resulting feedback can drive powerful winds that contribute significantly to the amount and metallicity of hot gas, both of which increase the X-ray luminosity. In terms of their stellar kinematics and structural properties, the merger remnants in our simulations resemble elliptical galaxies. We find that the X-ray luminosities of the remnants with B-band luminosities in the range L_B ~ 10^10 - 10^11 Lsun are consistent with observations, while remnants with smaller or larger masses are underluminous in X-rays. Moreover, because the majority of the merger remnants are broadly consistent with the observed scaling relations between temperature, B-band luminosity and X-ray luminosity we conclude that major mergers are a viable mechanism for producing the X-ray halos of large, luminous elliptical galaxies.