  We search for photometric variability in more than 23,000 known and candidate
white dwarfs, the largest ultraviolet survey compiled for a single study of
white dwarfs. We use gPhoton, a publicly available calibration/reduction
pipeline, to generate time-series photometry of white dwarfs 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 (< 30 min), we identify intrinsic variability in sources as
faint as Gaia G = 20 mag. With our ranking algorithm, we identify 49 new
variable white dwarfs (WDs) in archival GALEX observations. We detect 41 new
pulsators: 37 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%.
