In this paper, a generalized droop control (GDC) is proposed for a grid-supporting inverter based on a comparison between traditional droop control and virtual synchronous generator (VSG) control. Both the traditional droop control and VSG control have their own advantages, but neither traditional droop control nor VSG control can meet the demand for different dynamic characteristics in grid-connected (GC) and stand-alone (SA) modes at the same time. Rather than using a proportional controller with a low-pass filter, as in a traditional droop control, or fully mimicking the conventional synchronous generator parameters in a VSG control, the active power control loop of the GDC can be designed flexibly to adapt to different requirements. With a well-designed controller, the GDC can achieve satisfactory control performance; unlike a traditional droop control, it can provide virtual inertia and damping properties in SA mode; unlike a VSG control, the output active power of an inverter with GDC can follow changing references quickly and accurately, without large overshoot or oscillation in the GC mode. Moreover, given specific controller parameters, the GDC can function as both a traditional droop control and a VSG control. The GDC's controller parameter design is more intuitive and flexible, and this paper provides a distinct design process. Finally, the effectiveness of the proposed control method is validated by the simulation and experimental results.