Teachey et al. recently reported the detection of a candidate exomoon, tentatively designated Kepler-1625b I, around a giant planet in the Kepler field. The candidate exomoon would be about the size and mass of Neptune, considerably larger than any moon in our solar system, and if confirmed, would be the first in a new class of giant moons or binary planets. Motivated by the large mass ratio in the Kepler-1625b planet and satellite system, we investigate the detectability of similarly massive exomoons around directly imaged exoplanets via Doppler spectroscopy. The candidate moon around Kepler-1625b would induce a radial velocity (RV) signal of about 200 on its host planet, large enough that similar moons around directly imaged planets orbiting bright, nearby stars might be detected with current or next generation instrumentation. In addition to searching for exomoons, an RV survey of directly imaged planets could reveal the orientations of the planets' spin axes, making it possible to identify Uranus analogs.