The rapid expansion of low Earth orbit (LEO) satellite constellations has increased the tension between heterogeneous service demands and the limited capacity of intersatellite links (ISLs). In particular, meeting the divergent quality of service (QoS) requirements for ultrareliable low-latency communication (URLLC) and enhanced mobile broadband (eMBB) services poses significant challenges. To address these issues, we propose Q-RAP, which is an SDN-enabled, QoS-aware routing algorithm for LEO satellite megaconstellations. By combining multidimensional LP-based path optimization with a greedy partitioning scheme, Q-RAP satisfies the URLLC and eMBB requirements simultaneously. Extensive experiments on a joint STK–OpenStack simulation platform that emulates a Starlink constellation of 352 satellites demonstrate that the SDN-enabled Q-RAP reduces the end-to-end URLLC latency by up to 14.1% and shortens the routing convergence time by 26.0–28.6% compared with the state-of-the-art StarCure algorithm while incurring a bandwidth-efficiency penalty of less than 3%. Additionally, Q-RAP can reconfigure routes within 2 s under large-scale failure events, which highlights its robustness and scalability for next-generation satellite internet networks.