The Nancy Grace Roman Space Telescope (Roman) will perform a Galactic Exoplanet Survey (RGES) to discover bound exoplanets with semi-major axes greater than 1 au using gravitational microlensing. Roman will even be sensitive to planetary mass objects that are not gravitationally bound to any host star. Such free-floating planetary mass objects (FFPs) will be detected as isolated microlensing events with timescales shorter than a few days. A measurement of the abundance and mass function of FFPs is a powerful diagnostic of the formation and evolution of planetary systems, as well as the physics of the formation of isolated objects via direct collapse. We show that Roman will be sensitive to FFP lenses that have masses from that of Mars ($0.1 M_⊕$) to gas giants ($M\gtrsim100M_⊕$) as isolated lensing events with timescales from a few hours to several tens of days, respectively. We investigate the impact of the detection criteria on the survey, especially in the presence of finite-source effects for low-mass lenses. The number of detections will depend on the abundance of such FFPs as a function of mass, which is at present poorly constrained. Assuming that FFPs follow the fiducial mass function of cold, bound planets adapted from Cassan et al. (2012), we estimate that Roman will detect $∼250$ FFPs with masses down to that of Mars (including $∼ 60$ with masses $\le M_⊕$). We also predict that Roman will improve the upper limits on FFP populations by at least an order of magnitude compared to currently-existing constraints.