Distributed static compensators (DSTATCOMs) are grid-connected power electronic equipment dedicated to compensating reactive power as well as improving voltage regulation in distribution networks. They eclipse conventional compensation approaches, such as capacitor banks, in terms of flexibility and effectiveness. Despite their identified advantages, STATCOMs with voltage droop are subject to weak grid-induced stability problems, as first revealed by this paper. Specifically, the voltage droop controller that couples the amplitude of point of common coupling (PCC) voltages to the reactive current reference creates a local control loop. Such a loop greatly deteriorates system stability in weak grids, which feature large and variable grid impedances. To address such stability problems, we propose a novel virtual resistance control scheme, which improves system stability through mitigation of local control loop gains in the low-frequency band. Experimental results obtained from a DSTATCOM prototype clearly demonstrate the correctness of stability analysis and the effectiveness of stability improvement.