A recirculating superconducting linear accelerator with the advantage of both straight and circular accelerator has been demonstrated with relativistic electron beams. The acceleration concept of a recirculating proton beam was recently proposed [J. Qiang, Nucl. Instrum. Methods Phys. Res., Sect. A 795, 77 (2015)] and is currently under study. In order to further support the concept, the beam dynamics study on a recirculating proton linear accelerator has to be carried out. In this paper, we study the feasibility of a two-pass recirculating proton linear accelerator through the direct numerical beam dynamics design optimization and the start-to-end simulation. This study shows that the two-pass simultaneous focusing without particle losses is attainable including fully 3D space-charge effects through the entire accelerator system.