While the majority of massive stars have a stellar companion, most pulsars appear to be isolated. Taken at face value, this suggests that most massive binaries break apart due to strong natal kicks received in supernova explosions. However, the observed binary fraction can still be subject to strong selection effects, as monitoring of newly discovered pulsars is rarely carried out for long enough to conclusively rule out multiplicity. Here, we use the second Gaia data release to search for companions to 1534 rotation-powered pulsars with positions known to better than 0.5arcsec. We find 22 matches to known pulsars, including 1 not reported elsewhere, and 8 new possible companions to young pulsars. We examine the photometric and kinematic properties of these systems and provide empirical relations for identifying Gaia sources with potential millisecond pulsar companions. Our results confirm that the observed multiplicity fraction is small. However, we show that the number of binaries below the sensitivity of Gaia and radio timing in our sample could still be significantly higher. We constrain the binary fraction of young pulsars to be f^true^_young_<5.3(8.3) per cent under realistic (conservative) assumptions for the binary properties and current sensitivity thresholds. For massive stars (>10M_{sun}_) in particular, we find f^true^_OB_<3.7 per cent, which sets a firm independent upper limit on the Galactic neutron star merger rate, <=7.2x10^-4^/yr. Ongoing and future projects, such as the CHIME/pulsar program, MeerTime, HIRAX, and ultimately the SKA, will significantly improve these constraints in the future.