Islanded microgrids have low real and reactive power generation capacity and low inertia. This makes them susceptible to large frequency and voltage deviations, which deteriorate power quality and can cause frequency or voltage collapse. Grid-supporting battery energy storage systems are a possible solution as they are able to respond quickly to changes of their real and reactive power set-points. In this paper, a data-driven grid-supporting control system for battery energy storage systems, which requires no changes to the inverters inner real and reactive power control loops compared with a conventional grid-supporting inverter, is proposed. Tuning the data-driven controller does not require a dynamic model of the microgrid. Instead, the frequency response of the microgrid is identified and used directly to optimally tune the controller for $H_\infty$ performance and robustness criteria. The performance of the data-driven controller is verified through real-time software-in-the-loop electromagnetic-transient simulation, where it is compared with an inverse-droop controller and is shown to significantly reduce voltage and frequency deviations.