An equivalent definition of entropic Ricci curvature on discrete spaces was given in terms of the global gradient estimate. With a particular choice of the density function $ρ$, we obtain a localized gradient estimate, which in turns allow us to derive a Bonnet-Myers type diameter bound for graphs with positive entropic Ricci curvature. However, the case of the hypercubes indicates that the bound may be not optimal (where $θ$ is chosen to be logarithmic mean by default). If $θ$ is arithmetic mean, the Bakry-Émery criterion can be recovered and the diameter bound is optimal as it can be attained by the hypercubes.