How the interacting electronic states and phases of layered transition-metal dichalcogenides evolve when thinned to the single-layer limit is a key open question in the study of two-dimensional materials. Here, we use angle-resolved photoemission to investigate the electronic structure of monolayer VSe₂ grown on bilayer graphene/SiC. While the global electronic structure is similar to that of bulk VSe₂, we show that, for the monolayer, pronounced energy gaps develop over the entire Fermi surface with decreasing temperature below T_c = 140 ± 5 K, concomitant with the emergence of charge-order superstructures evident in low-energy electron diffraction. These observations point to a charge-density wave instability in the monolayer that is strongly enhanced over that of the bulk. Moreover, our measurements of both the electronic structure and of X-ray magnetic circular dichroism reveal no signatures of a ferromagnetic ordering, in contrast to the results of a recent experimental study as well as expectations from density functional theory. Our study thus points to a delicate balance that can be realized between competing interacting states and phases in monolayer transition-metal dichalcogenides.