We investigate a novel downlink multicarrier direct-sequence code division multiple access (MC-DS-CDMA) resource allocation scheme in the context of low earth orbit (LEO) satellite-ground integrated networks (SGINs). In contrast to the existing MC-DS-CDMA works which mainly focus on delay-tolerant services in terrestrial networks, we consider the heterogeneous delay traffic and take the unique characteristics of LEO satellite systems into account. Specifically, we exploit the channel information, the delay requirement, the buffer state, and the visible time of users to construct a utility function and formulate the resource allocation optimization problem in presence of co-channel interferences (CCI) in LEO satellite-ground heterogeneous systems. We transform the original nonconvex optimization problem into two convex ones and use the Lagrange dual decomposition method to derive the solution. We also design an efficient algorithm dynamically scheduling subcarriers, codes, and transmission power of MC-DS-CDMA. The simulation results confirm the superiority of our work in terms of lower average delay and higher overall throughput.