Integrating monoclinic gallium oxide (β-Ga₂O₃) with two-dimensional (2D) hexagonal boron nitride (h-BN) into heterostructures is of significant importance for achieving high-power device applications. The 2D-material-assisted epitaxy provides a straightforward integration method for fabricating β-Ga₂O₃/h-BN vertical heterostructures. In this work, the β-Ga₂O₃ films were deposited on both polycrystalline and single-crystalline h-BN layers with different thicknesses, and two growth modes of β-Ga₂O₃ films on h-BN, remote epitaxy, and van der Waals (vdW) epitaxy, were investigated. The results show that the potential of the sapphire substrate can penetrate the monolayer and bilayer h-BN to obtain the remote epitaxy of β-Ga₂O₃ films, regardless of the crystallinity of h-BN. The vdW epitaxy of β-Ga₂O₃ film can be realized on the monocrystalline h-BN substrate. Compared with the conventional and remote epitaxial β-Ga₂O₃ films on sapphire substrate, the vdW epitaxial β-Ga₂O₃ films on the single-crystalline h-BN substrate exhibit higher crystallinity. This work indicates that the 2D-material-assisted epitaxy provides a feasible scheme for the heterogeneous integration of β-Ga₂O₃ films.