We present a gyrokinetic theory of long-term collisionless damping of a self-generated monopolar E × B vortex flow in a tokamak magnetic island. By an explicit analytic calculation in the central island region, we show that the magnetic precession-induced coupling of the monopolar vortex to the island geodesic acoustic mode (IGAM) results in long-term oscillatory damping. Our theory reveals that IGAM is qualitatively different from both GAM and a sound wave. The IGAM signal can be utilized as an indicator of the turbulence invasion into the tokamak magnetic island.