We investigate the stability of variable-density two-dimensional isolated vortices in the frame of incompressible mixing under negligible gravity. The focus on a single vortex flow stands as a first step towards vortex interactions and turbulent mixing. From heuristic arguments developed on a perturbed barotropic vortex, we find that high-density vortices are subject to a Rayleigh–Taylor instability. The basic mechanism relies on baroclinic vorticity generation when the density gradient is misaligned with the centripetal acceleration field. For Gaussian radial distributions of vorticity and density, the intensity of the baroclinic torque due to isopycnic deformation is shown to increase with the ratio $\delta/\delta_\rho$ of the vorticity radius to the density radius. Concentration of mass near the vortex core is confirmed to promote the instability by the use of an inviscid linear stability analysis. We measure the amplification rate for the favoured azimuthal wavenumbers $m\,{=}\,2,3$ on the whole range of positive density contrasts between the core and the surroundings. The separate influence of the density-contrast and the radius ratio is detailed for modes up to $m\,{=}\,6$. For growing azimuthal wavenumbers, the two-dimensional structure of the eigenmode concentrates on a ring of narrowing radial extent centred on the radius of maximum density gradient. The instability of the isolated high-density vortex is then explored beyond the linear stage based on high-Reynolds-number numerical simulations for modes $m\,{=}\,2,3$ and a moderate density contrast $C_{\rho}\,{=}\,0.5$. Secondary roll-ups are seen to emerge from the nonlinear evolution of the vorticity and density fields. The transition towards $m$ smaller vortices involves vorticity exchange between initially-rotating dense fluid particles and the irrotational less-dense medium. It is shown that baroclinic enstrophy production is associated with the centrifugal mass ejection away from the vortex centre.