We compare high-resolution ultraviolet spectra of the Sun and thirteen solar-mass main-sequence stars with different rotational periods that serve as proxies for their different ages and magnetic field structures. In this, the second paper in the series, we study the dependence of ultraviolet emission-line centroid velocities on stellar rotation period, as rotation rates decrease from that of the Pleiades star HII314 (P_rot_=1.47days) to {alpha} Cen A (P_rot_=28days). Our stellar sample of F9 V to G5 V stars consists of six stars observed with the Cosmic Origins Spectrograph on the Hubble Space Telescope (HST) and eight stars observed with the Space Telescope Imaging Spectrograph on HST. We find a systematic trend of increasing redshift with more rapid rotation (decreasing rotation period) that is similar to the increase in line redshift between quiet and plage regions on the Sun. The fastest-rotating solar-mass star in our study, HII314, shows significantly enhanced redshifts at all temperatures above logT=4.6, including the corona, which is very different from the redshift pattern observed in the more slowly rotating stars. This difference in the redshift pattern suggests that a qualitative change in the magnetic-heating process occurs near P_rot_=2days. We propose that HII314 is an example of a solar-mass star with a magnetic heating rate too large for the physical processes responsible for the redshift pattern to operate in the same way as for the more slowly rotating stars. HII314 may therefore lie above the high activity end of the set of solar-like phenomena that is often called the "solar-stellar connection."