In this paper, we investigate a user pairing problem in power domain non-orthogonal multiple access (NOMA) scheme-aided satellite networks. In the considered scenario, different satellite applications are assumed with various delay quality-of-service (QoS) requirements, and the concept of effective capacity is employed to characterize the effect of delay QoS limitations on achieved performance. Based on this, our objective was to select users to form a NOMA user pair and utilize resource efficiently. To this end, a power allocation coefficient was firstly obtained by ensuring that the achieved capacity of users with sensitive delay QoS requirements was not less than that achieved with an orthogonal multiple access (OMA) scheme. Then, considering that user selection in a delay-limited NOMA-based satellite network is intractable and non-convex, a deep reinforcement learning (DRL) algorithm was employed for dynamic user selection. Specifically, channel conditions and delay QoS requirements of users were carefully selected as state, and a DRL algorithm was used to search for the optimal user who could achieve the maximum performance with the power allocation factor, to pair with the delay QoS-sensitive user to form a NOMA user pair for each state. Simulation results are provided to demonstrate that the proposed DRL-based user selection scheme can output the optimal action in each time slot and, thus, provide superior performance than that achieved with a random selection strategy and OMA scheme.