Observing the significance of spectrally-efficient secure non-orthogonal multiple access (NOMA), this paper proposes a novel quality of service (QoS) aware secure NOMA protocol that maximizes secrecy fairness among untrusted users. Considering a base station (BS) and two users, a novel decoding order is designed that provides security to both users. With the objective of ensuring secrecy fairness between users, while satisfying their QoS requirements under BS transmit power budget constraint, we explore the problem of minimizing the maximum secrecy outage probability (SOP). Closed-form expression of pair outage probability (POP) and optimal power allocation (PA) minimizing POP are obtained. To analyze secrecy performance, analytical expressions of SOP for both users are derived, and individual SOP minimization problems are solved using concept of generalized-convexity. High signal-to-noise ratio approximation of SOP and asymptotically optimized solution minimizing this approximation is also found. Furthermore, a global-optimal solution from secrecy fairness standpoint is obtained at low computational complexity, and tight approximation is derived to get analytical insights. Numerical results present useful insights on globally optimized PA which ensure secrecy fairness and provide performance gain of about 55.12%, 69.30%, and 19.11%, respectively, compared to fixed PA and individual users' optimal PAs. Finally, a tradeoff between secrecy fairness performance and QoS demands is presented.