This paper proposes a green computing strategy for low Earth orbit (LEO) satellite networks (LSNs), addressing energy efficiency and delay optimization in dynamic and energy-constrained environments. By integrating a Markov Decision Process (MDP) with a Double Deep Q-Network (Double DQN) and introducing the Energy–Delay Ratio (EDR) metric, this study effectively quantifies and balances energy savings with delay costs. Simulations demonstrate significant energy savings, with reductions of up to 47.87% under low business volumes, accompanied by a minimal delay increase of only 0.0161 s. For medium business volumes, energy savings reach 26.75%, with a delay increase of 0.0189 s, while high business volumes achieve a 4.36% energy reduction and a delay increase of 0.0299 s. These results highlight the proposed strategy’s ability to effectively balance energy efficiency and delay, showcasing its adaptability and suitability for sustainable operations in LEO satellite networks under varying traffic loads.