The linear stability of unbounded strained vortices in a stably stratified rotating fluid is investigated theoretically. The problem is reduced to a Matrix–Floquet problem, which is solved numerically to determine the stability characteristics. The Coriolis force and the buoyancy force suppress the subharmonic elliptical instability of cyclonic and weak anticyclonic vortices, whereas enhances that of strong anticyclonic vortices. The fundamental and superharmonic instability modes occur, in addition. They are due to higher‐order resonance. The growth rate of each instability shows complicated dependence on the parametersN(the normalized Brunt–Va¨isa¨la¨ frequency) andR0(the Rossby number: defined inversely as usual), if their values are small. It decreases as the background rotation rate becomes larger and as the stratification becomes stronger. The instability mode whose order of resonance is less than Min(N,2‖1+R0‖) is inhibited.