The communicability distance (Estrada, Lin. Alg. Appl. 2012, 436, 4317) is a useful metric to characterize alternative navigational routes in graphs. Here we prove that it is the resistance distance between a pair of nodes in a weighted graph. We extend this result and prove that every nonsingular Euclidean distance matrix is the resistance distance matrix of a weighted graph. We briefly analyze some mathematical properties of the communicability Laplacian matrix which emerges from the current analysis.