The Greenland ice sheet is expected lose at least 90% of its current volume if ice sheet summer temperatures warm by around 1.8 °C above pre‐industrial. Geoengineering by stratospheric sulfate aerosol injection might slow Greenland ice sheet melting and sea level rise by reducing summer temperature and insolation; however, such schemes could also reduce precipitation and affect large‐scale climate drivers such as the Atlantic Meridional Over‐turning Circulation (AMOC). Earlier work found that AMOC increased under geoengineering and that might lead to greater mass loss from Greenland than under greenhouse gas forcing alone. We simulated Greenland ice sheet climates using four Earth system models running the stratospheric sulfate aerosol injection experiment GeoMIP G4 and the CMIP RCP4.5 and RCP8.5 greenhouse gas scenarios that were then used to drive the surface energy and mass balance model, SEMIC. Simulated runoff is 20% lower under G4 than RCP4.5, while under RCP8.5 it is 17% higher. The mechanism is through increased Arctic sea ice concentration and reduced humidity leading to surface cooling of the ablation zone. Reduced absorption of outgoing longwave radiation caused by hydrological cycle weakening dominates associated decreases in precipitation under geoengineering and stronger AMOC than under RCP4.5. An ice dynamics model simulates 15% lower ice losses under G4 than RCP4.5. Thus, total sea level rise by 2070 from the Greenland ice sheet under G4 geoengineering is about 15–20% lower than under the RCP4.5 scenario.