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This paper proposes a fully distributed approximate dynamic programming (FD-ADP) algorithm framework for real-time economic dispatch of a microgrid (MG). Firstly, a modified distributed consensus-based algorithm (DCBA) framework with power restriction factors is proposed to distributedly obtain the real-time ED decision of MG containing distributed energy resources with linear cost functions, which improves the convergence stability compared with the traditional DCBA framework. Then, combined with the approximate dynamic programming (ADP) theory, a FD-ADP algorithm is proposed to deal with data uncertainties. To this end, the piece-wise linear (PWL) function is applied to approximate the value function, and a distributed slope update approach is proposed to embed empirical knowledge into the value function while ensuring the distributed property in off-line training. With the well-trained PWL function assisting in decision, the distributed real-time ED strategy of MG with approximate global optimality is obtained by the proposed FD-ADP algorithm during the entire schedule horizon (e.g., 24 hours). Finally, case studies on the modified IEEE 6-bus MG system and IEEE 123-bus MG system demonstrate the superiority of solution convergence, validity of distributed training, and optimality of distributed schedule of the proposed FD-ADP algorithm.