The recirculating superconducting proton linac has the advantage of reducing the number of cavities in the accelerator and the corresponding construction and operational costs. Beam dynamics simulations were done recently in a double-pass recirculating proton linac using a single proton beam bunch. For continuous wave (cw) operation, the high-energy proton bunch during the second pass through the linac will overtake and collide with the low-energy bunch during the first pass at a number of locations of the linac. These collisions might cause proton bunch emittance growth and beam quality degradation. In this paper, we study the collisional effects due to Coulomb space-charge forces between the high-energy bunch and the low-energy bunch. Our results suggest that these effects on the proton beam quality would be small and might not cause significant emittance growth or beam blowup through the linac. A 10 mA, 500 MeV cw double-pass proton linac is feasible without using extra hardware for phase synchronization.