Heterogeneous engineering of two-dimensional layered materials, including metallic graphene and semiconducting transition metal dichalcogenides, presents an exciting opportunity to produce highly tunable electronic and optoelectronic systems. In order to engineer pristine layers and their interfaces, epitaxial growth of such heterostructures is required. We report the direct growth of crystalline, monolayer tungsten diselenide (WSe2) on epitaxial graphene (EG) grown from silicon carbide. Raman spectroscopy, photoluminescence, and scanning tunneling microscopy confirm high-quality WSe2 monolayers, whereas transmission electron microscopy shows an atomically sharp interface, and low energy electron diffraction confirms near perfect orientation between WSe2 and EG. Vertical transport measurements across the WSe2/EG heterostructure provides evidence that an additional barrier to carrier transport beyond the expected WSe2/EG band offset exists due to the interlayer gap, which is supported by theoretical local density of states (LDOS) calculations using self-consistent density functional theory (DFT) and nonequilibrium Green's function (NEGF).