  After large galaxies merge, their central supermassive black holes are
expected to form binary systems whose orbital motion generates a gravitational
wave background (GWB) at nanohertz frequencies. Searches for this background
utilize pulsar timing arrays, which perform long-term monitoring of millisecond
pulsars (MSPs) at radio wavelengths. We use 12.5 years of Fermi Large Area
Telescope data to form a gamma-ray pulsar timing array. Results from 35 bright
gamma-ray pulsars place a 95\% credible limit on the GWB characteristic strain
of $1.0\times10^{-14}$ at 1 yr$^{-1}$, which scales as the observing time span
$t_{\mathrm{obs}}^{-13/6}$. This direct measurement provides an independent
probe of the GWB while offering a check on radio noise models.
