When a droplet with a higher density falls in a miscible solution, the droplet deforms and breaks up. The instability of a vortex ring, formed by droplet deformation during the falling process, causes the breakup. To determine the origin of the instability, the wavelengths and thicknesses of the vortex rings are investigated at the time when the instability occurs. The experimental results are almost in agreement with the calculated results for the Rayleigh-Taylor instability using the thickness of a higher-density solution. Furthermore, we performed simulations considering the torus shapes and circulations of the vortex ring. The simulations provided patterns similar to those observed experimentally for the breakup process, and showed that the circulations suppress the instability of the vortex ring. These results imply that the Rayleigh-Taylor instability plays a dominant role in the instability of vortex rings.