The stability of ionospheric O+-H+ outflows accelerated by a nonambipolar parallel electric field is considered under conditions where (1) the ion motion initially developes adiabatically and (2) the ambient plasma is vertically stratified with an effective temperature that increases with altitude. Such conditions are expected near the bottom of the auroral acceleration region where ion and electron streaming instabilities first develop. It is shown for a particular equilibrium profile that the differentially accelerated ion flows become unstable within about 100 km from their entry point in the acceleration region. At O+/H+ density ratios less than about 9, the instability is dominated by a violent H+-O+ two-stream interaction which couples the O+ and H+ acoustic modes, and which mediates a transition to nonadiabatic acceleration. At high altitudes and/or larger O+/H+ density ratios, a much weaker resonant instability exists, which is driven by the relative drift between electrons and O+ or H+ ions. The results suggest that the H+-O+ two-stream instability may be a viable mechanism for heating upflowing auroral ions.