We search for photometric variability in more than 23000 known and candidate white dwarfs (WDs), the largest ultraviolet survey compiled for a single study of WDs. We use GPHOTON, a publicly available calibration/reduction pipeline, to generate time-series photometry of WDs observed by GALEX. By implementing a system of weighted metrics, we select sources with variability due to pulsations and eclipses. Although GALEX observations have short baselines (=<30min), we identify intrinsic variability in sources as faint as Gaia G=20mag. With our ranking algorithm, we identify 48 new variable WDs in archival GALEX observations. We detect 40 new pulsators: 36 have hydrogen-dominated atmospheres (DAVs), including one possible massive DAV, and four are helium-dominated pulsators (DBVs). We also detect eight new eclipsing systems; five are new discoveries, and three were previously known spectroscopic binaries. We perform synthetic injections of the light curve of WD 1145+017, a system with known transiting debris, to test our ability to recover similar systems. We find that the 3{sigma} maximum occurrence rate of WD 1145+017-like transiting objects is =<0.5 per cent.