In this paper, a new discrete-time data-driven distributed learning control strategy for frequency/voltage regulation and active/reactive power sharing of islanded microgrids is proposed. Instead of using the static droop relationship and the conventional primary-secondary hierarchical control structure, a new control framework is adopted and a neural network is used to learn the control law. The neural network is tuned online using the operational system input/output data with no training phase. As a result, the transient performance of microgrids is improved and a remarkable plug-and-play capability is also achieved. Moreover, the stability of the closed-loop system is analyzed through the Lyapunov approach, where the interactions between different distributed energy resources are considered. The effectiveness of the proposed method is demonstrated by real-time hardware-in-the-loop experiment of a typical microgrid.