Machine learning (ML) is revolutionizing protein structural analysis, including an important subproblem of predicting protein residue contact maps, i.e., which amino-acid residues are in close spatial proximity given the amino-acid sequence of a protein. Despite recent progresses in ML-based protein contact prediction, predicting contacts with a wide range of distances (commonly classified into short-, medium- and long-range contacts) remains a challenge. Here, we propose a multiscale graph neural network (GNN) based approach taking a cue from multiscale physics simulations, in which a standard pipeline involving a recurrent neural network (RNN) is augmented with three GNNs to refine predictive capability for short-, medium- and long-range residue contacts, respectively. Test results on the ProteinNet dataset show improved accuracy for contacts of all ranges using the proposed multiscale RNN+GNN approach over the conventional approach, including the most challenging case of long-range contact prediction.