We assess the accuracy and spatial resolution of the Surface Water and Ocean Topography (SWOT) swath altimeter for measuring marine gravity anomalies. The analysis is performed at the Foundation Seamounts in the South Pacific where we developed a highly accurate gravity field by combining the long-wavelength (> 40 km) gravity field derived from previous nadir altimeters with the shorter wavelength gravity field from the seafloor topography as constrained by the ship gravity. In this region, the slope of the ocean variability is 50-100 times smaller than the gravity/slope signal of the seamounts so can be ignored in the analysis. Each SWOT cycle can deliver gravity anomaly/SSS with an accuracy of 2.6 mGal/μrad and a spatial resolution of 14 km, with accuracy diminishing when significant wave height (SWH) exceeds ~6 meters. Averaging repeated SWOT measurements improves the accuracy and resolution. For example, we expect that averaging just 10 repeats (7 months) results in accuracy/resolution that matches the best marine gravity maps based on 230 months of nadir altimetry. With a mission lasting over a year, SWOT promises a substantial leap in marine gravity accuracy and resolution, uncovering previously uncharted details of the seafloor, including thousands of uncharted seamounts.