In this paper, a data-driven robust distributed control strategy is proposed for cooperative voltage regulation of the direct-current (DC) microgrids such that the output voltages of all distributed generation units (DGUs) track the specified reference signal. Unlike in decentralized control where all DGUs need to know the information of the reference signal, each DGU only needs to obtain the information of its neighbors in the proposed distributed control strategy. In the presence of the unknown parameters of DGUs, a data-driven robust control method is proposed to achieve cooperative voltage regulation under the available data affected by noise. It is shown that the cooperative voltage regulation problem can be transformed into a local stabilization problem based on the proposed distributed control protocol. To address the local stabilization problem, the noisy data from local DGU and the constructed auxiliary system is collected, and then we develop a data-driven control framework that guarantees robust closed-loop stability under bounded noise. We formulate a data-dependent linear matrix inequality (LMI) based on ${\mathcal{D}}_Q$-stabilization theory. This LMI enforces all eigenvalues of the closed-loop system to lie within a specified region inside the unit disk, thereby quantifying robustness through guaranteed stability margins when the collected data is corrupted by noise. The resulting gain matrices of the distributed control protocol are computed directly from noisy-data-dependent LMI without requiring any explicit model knowledge. Several simulation results are provided to show the effectiveness and robustness of the proposed methods.