We apply a surface flux transport model developed for the Sun to reconstruct the stellar activity-rotation relationship, LX/Lbol versus Ro, observed for unsaturated cool stars (Rossby numbers Ro ≳ 0.1). This empirical flux transport model incorporates modulations of magnetic flux strength consistent with observed solar activity cycles, as well as surface flux dynamics consistent with observed and modeled stellar relationships. We find that for stellar flux models corresponding to a range of 0.1 ≲ (Ro/RoSun) ≲ 1.2, the LX/Lbol versus Ro relation matches well the power-law behavior observed in the unsaturated regime of cool stars. Additionally, the magnetic activity cycles captured by the stellar simulations produce a spread about the power-law relation consistent with that observed in cool star populations, indicating that the observed spread may be caused by intrinsic variations resulting from cyclic stellar behavior. The success of our flux transport modeling in reproducing the observed activity relationship across a wide range of late-F, G, K, and M stars suggests that the photospheric magnetic fields of all unsaturated cool stars exhibit similar flux emergence and surface dynamic behavior, and may hint at possible similarities in stellar dynamo action across a broad range of stellar types.