This paper considers satellite communication networks where each satellite terminal is equipped with energy harvesting (EH) devices to supply energy continuously, and randomly transmits bursty packets to a geostationary satellite over a shared wireless channel. Packet replicas combined with a successive iteration cancellation scheme can reduce the negative impact of packet collisions but consume more energy. Hence, appropriate energy management policies are required to mitigate the adverse effect of energy outages. Although centralized access schemes can provide better performance on the networks’ throughput, they expend extra signallings to allocate the resources, which leads to non-negligible communication latencies, especially for the satellite communication networks. In order to reduce the communication overhead and delay, a distributed random access (RA) scheme considering the energy constraints is studied. Each EH satellite terminal (EH-ST) decides whether to transmit the packet and how many replicas are transmitted according to its local energy and EH rates to maximize the average long-term network throughput. Owing to the nonconvexity of this problem, we adopted a game theoretic method to approximate the optimal solution. By forcing all the EH-STs to employ the same policy, we characterized and proved the existence and uniqueness of the symmetric Nash equilibrium (NE) of the game. Moreover, an efficient algorithm is proposed to calculate the symmetric NE by combining a policy iteration algorithm and the bisection method. The performance of the proposed RA scheme was investigated via numerous simulations. Simulation results showed that the proposed RA scheme is applicable to the EH devices in the future low-cost interactive satellite communication system.