This paper proposes an adaptive and distributed secondary voltage control for microgrids with inverter-based distributed generators (DG). The proposed control is fully adaptive and does not require the information of DG parameters. Neural networks are used to compensate for the uncertainties caused by the unknown dynamics of DGs. The controller structure is fully distributed such that each DG only requires its own information and the information of its neighbors on the communication network. Therefore, this secondary control is associated with a sparse communication network. The effectiveness of the proposed methodology is verified for different loading, outage, and islanding scenarios, as well as variable communication structures in a microgrid setup.