GaSe is an important member of the post-transition metal chalcogenide family and is an emerging two-dimensional (2D) semiconductor material. Because it is a van der Waals (vdW) material, it can be fabricated into atomic-scale ultrathin films, making it suitable for the preparation of compact, heterostructure devices. In addition, GaSe possesses unusual optical and electronic properties, such as a shift from an indirect-bandgap single-layer film to a direct-bandgap bulk material, rare intrinsic p-type conduction, and nonlinear optical behaviors. These properties make GaSe an appealing candidate for the fabrication of field-effect transistors, photodetectors, and photovoltaics. However, the wafer-scale production of pure GaSe single crystal thin films remains challenging. This study develops an approach for the direct growth of GaSe thin films on GaAs substrates using molecular beam epitaxy. It yields smooth thin GaSe films with a {\gamma}'-configuration, a recently-proposed novel polymorph. We analyze the formation mechanism of {\gamma}'-GaSe using density functional theory, finding that this polymorph is stabilized by Ga vacancies. Finally, we investigate the growth conditions of GaSe, providing valuable insights for exploring 2D/3D quasi-vdW epitaxial growth.