Within the framework of nonrelativistic effective field theory, we calculate the radiative decay widths of the process $X(3872)\to D\bar{D}\gamma$ while taking into account the $D\bar{D}$ final-state interactions via resonance saturation. The $X(3872)$ is treated as a $D^*\bar{D} +\rm c.c.$ bound state with $J^{PC}=1^{++}$. For a representative mixing angle $\theta =\pi /4$, our numerical calculations predict that the tree-level partial decay width for $X(3872)\to D^0{\bar{D}}^0\gamma$ can reach approximately $7.3$ keV, while that for $X(3872)\to D^{+}{D}^{-}\gamma$ is less than $1$ keV. It is found that the contribution of the triangle loop diagrams to the total decay width of $X(3872)\to D^0{\bar{D}}^0\gamma$ is very small, amounting to less than $1\%$ of the tree-level result. For the $X(3872)D^+ D^-$ decay, the contribution from the triangle loop diagrams is relatively more significant, enhancing the total decay width by roughly $1.3\%$ relative to the tree-level result. We expect that the present predictions based on the molecular picture of the $X(3872)$ can be tested by future experiments.