We present a local stability analysis of an idealized model of stratified vortices that appear in geophysical settings. The base flow comprises an axisymmetric vortex with background rotation and an out-of-plane stable stratification, and a radial stratification in thermal wind balance with the out-of-plane momentum gradient. Solving the local stability equations along fluid particle trajectories in the base flow, the dependence of short-wavelength instabilities on Schmidt number Sc is studied, in the presence of curvature effects. In the diffusion-free limit, the well-known symmetric instability is recovered. In the viscous, double-diffusive regime, instability characteristics are shown to depend on three non-dimensional parameters (including Sc), and two different instabilities are identified: 1. a monotonic instability (same as symmetric instability at Sc = 1), and 2. an oscillatory instability (absent at Sc = 1). Separating the base flow and perturbation characteristics, two each of base flow and perturbation parameters (apart from Sc) are identified, and the entire parameter space is explored for the aforementioned instabilities. In comparison to Sc = 1, monotonic and oscillatory instabilities are shown to significantly expand the instability region in the space of base flow parameters as Sc moves away from unity. Neutral stability boundaries on the plane of Sc and a modified gradient Richardson number are then identified for both these instabilities. We conclude with a discussion of curvature effects, and the likelihood of monotonic and oscillatory instabilities in typical oceanic settings.