The evolution of a counter-rotating vortex pair in a stably stratified fluid is investigated using direct numerical simulations. The study focuses on the short-wavelength elliptic instability occurring in this flow and the subsequent decay of the vortices. Depending on the level of stratification, as characterized by the Froude number which indicates the time scale of buoyancy to that of the instability, and the stage of evolution, stratification effects may significantly alter the behaviour of the flow. In the case of weak to moderate stratification, the elliptic instability develops qualitatively in the same manner as in unstratified fluid. The primary effect of stratification is to reduce the vortex separation distance which enhances the mutually induced strain. Consequently, the instability has an earlier onset and higher growth rate with increasing stratification. The behaviour is essentially described by linear stability theory for unstratified flow if the varying separation distance is taken into account. On the other hand, the final breakdown and decay of the flow may be greatly modified by stratification since buoyancy effects eventually emerge after sufficient time has elapsed. The decay is enhanced owing to additional mechanisms not present in unstratified flow. Secondary vertical vortex structures form between the primary vortices promoting fluid exchange in the transverse direction. Detrainment of fluid from the primary vortices by the generated baroclinic torque also contributes to the more rapid breakdown of the flow. In the case of strong stratification, in which the time scale of buoyancy is comparable to that of the instability, the flow is significantly altered. As a result of strong baroclinic torque, the primary vortices are brought together and detrainment occurs earlier. The associated reduction in radii of the vortices results in a higher axial wave mode and a more complex radial structure of the instability. Detrainment and mixing accelerate their decay. Late time evolution is dominated by the successive generation of alternate signed baroclinic torque which results in an oscillation of the total flow circulation at the buoyancy frequency.