We present the results of the first systematic search for spectroscopic binaries within the central 2 × 3 arcsec^2 around the supermassive black hole at the center of the Milky Way galaxy. This survey is based primarily on over a decade of adaptive optics-fed integral-field spectroscopy (R ∼ 4000), obtained as part of the Galactic Center Orbits Initiative at Keck Observatory, and it has a limiting K'-band magnitude of 15.8, which is at least 4 mag deeper than previous spectroscopic searches for binaries at larger radii within the central nuclear star cluster. From this primary data set, over 600 new radial velocities are extracted and reported, increasing by a factor of 3 the number of such measurements. We find no significant periodic signals in our sample of 28 stars, of which 16 are massive, young (main-sequence B) stars and 12 are low-mass, old (M and K giant) stars. Using Monte Carlo simulations, we derive upper limits on the intrinsic binary star fraction for the young star population at 47% (at 95% confidence) located ∼20 mpc from the black hole. The young star binary fraction is significantly lower than that observed in the field (70%). This result is consistent with a scenario in which the central supermassive black hole drives nearby stellar binaries to merge or be disrupted, and it may have important implications for the production of gravitational waves and hypervelocity stars.