Binary asteroids probe thermal-radiation effects on the main-belt asteroids' evolution. We discuss the possibility of detecting binary minor planet systems by the astrometric wobble of the center-of-light around the center-of-mass. This method enables the exploration of the phase-space of binary asteroids, which is difficult to explore using common detection techniques. We describe a forward model that projects the center-of-light position with respect to the center-of-mass, as it is seen by the observer. We study the performance of this method using simulated Gaia-like data. We apply the astrometric method to a subset of the Gaia DR2 Solar System catalog and find no significant evidence of binary asteroids. This is likely because the Gaia DR2 removed astrometric outliers, which in our case may be due to astrophysical signals. Applying this method to binary asteroid (4337) Arecibo, for which Gaia DR3 reported a possible astrometric signal with a period of P = 32.85+/-0.38 hr, reveals a possible 2.2-sigma solution with a period of 16.26 hr (about half the reported period). We find a small, marginally significant, excess of astrometric noise in the known binary asteroid population from Pravec et al. relative to the entire asteroid population in the Gaia DR2 Solar System catalog. We also discuss some caveats like precession and asteroid rotation.