Cyclone–anticyclone asymmetry in spontaneous gravity wave radiation from a co-rotating vortex pair is investigated in an $f$ -plane shallow water system. The far field of gravity waves is derived analytically by analogy with the theory of aeroacoustic sound wave radiation (Lighthill theory). In the derived form, the Earth’s rotation affects not only the propagation of gravity waves but also their source. While the results correspond to the theory of vortex sound in the limit of $f\rightarrow 0$ , there is an asymmetry in gravity wave radiation between cyclone pairs and anticyclone pairs for finite values of $f$ . Anticyclone pairs radiate gravity waves more intensely than cyclone pairs due to the effect of the Earth’s rotation. In addition, there is a local maximum of intensity of gravity waves from anticyclone pairs at an intermediate $f$ . To verify the analytical solution, a numerical simulation is also performed with a newly developed spectral method in an unbounded domain. The novelty of this method is the absence of wave reflection at the boundary due to a conformal mapping and a pseudo-hyperviscosity that acts like a sponge layer in the far field of waves. The numerical results are in excellent agreement with the analytical results even for finite values of $f$ for both cyclone pairs and anticyclone pairs.