Giant X-ray outbursts, with luminosities of about 10³⁷ erg s^–1, are observed roughly every five years from the nearby Be/pulsar binary 1A 0535+262. In this article, we present observations of the source with VERITAS at very high energies (VHEs;E >100 GeV) triggered by the X-ray outburst in 2009 December. The observations started shortly after the onset of the outburst and provided comprehensive coverage of the episode, as well as the 111 day binary orbit. No VHE emission is evident at any time. We also examined data from the contemporaneous observations of 1A 0535+262 with the Fermi/Large Area Telescope at high-energy photons (E > 0.1 GeV) and failed to detect the source at GeV energies. The X-ray continua measured with theSwift/X-Ray Telescope and the RXTE/PCA can be well described by the combination of blackbody and Comptonized emission from thermal electrons. Therefore, the gamma-ray and X-ray observations suggest the absence of a significant population of non-thermal particles in the system. This distinguishes 1A 0535+262 from those Be X-ray binaries (such as PSR B1259-63 and LS I +61°303) that have been detected at GeV-TeV energies. We discuss the implications of the results on theoretical models.