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In this paper, we develop data-driven closure/correction terms to increase the pressure and velocity accuracy of reduced order models (ROMs) for fluid flows. Specifically, we propose the first pressure-based data-driven variational multiscale ROM, in which we use the available data to construct closure/correction terms for both the momentum equation and the continuity equation. Our numerical investigation of the two-dimensional flow past a circular cylinder at Re=50000 in the marginally-resolved regime shows that the novel pressure data-driven variational multiscale ROM yields significantly more accurate velocity and pressure approximations than the standard ROM and, more importantly, than the original data-driven variational multiscale ROM (i.e., without pressure components). In particular, our numerical results show that adding the closure/correction term in the momentum equation significantly improves both the velocity and the pressure approximations, whereas adding the closure/correction term in the continuity equation improves only the pressure approximation.