After large galaxies merge, their central supermassive black holes are expected to form binary systems. Their orbital motion should generate a gravitational wave background (GWB) at nanohertz frequencies. Searches for this background use pulsar timing arrays, which perform long-term monitoring of millisecond pulsars at radio wavelengths. We used 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 x 10(-14) at a frequency of 1 year(-1). The sensitivity is expected to scale with t(obs), the observing time span, as t(obs)(-13/6). This direct measurement provides an independent probe of the GWB while offering a check on radio noise models.