The GAGG Radiation Instrument (GARI) is designed to space-qualify a compact, high-sensitivity gamma-ray spectrometer for astrophysical and defense applications and has completed over one year of operations on the International Space Station (ISS). The on-orbit activation of the GAGG crystal induced by the radiation background was measured. Characteristic gamma-ray lines present in the on-orbit spectra were compared to ground-based tests for identification. The radiation background, including the particle-induced internal activation of the crystal, affects the sensitivity of the instrument. We also show the degradation in the performance of the silicon photomultiplier (SiPM) readout (known to be sensitive to radiation damage). Results shown here will be useful in predicting the performance of larger instruments that use GAGG scintillator technology for gamma-ray spectroscopy.