In underwater acoustic networks (UANs), communication between nodes is susceptible to long propagation delays, limited energy, and channel conflicts, and traditional multi-access control (MAC) protocols cannot easily cope with these challenges. To enhance network throughput and balance channel allocation fairness and energy efficiency, this paper proposes a multi-objective optimization MAC protocol (MOMA-MAC) based on multi-agent reinforcement learning. MOMA-MAC utilizes a delay reward mechanism combined with the Multi-agent Proximal Policy Optimization Algorithm (MAPPO) to design a dual reward mechanism, which enables agents to adaptively collaborate and compete to optimize the use of network resources. According to experimental results, MOMA-MAC performs noticeably better than traditional MAC protocols and deep reinforcement learning-based methods in terms of throughput, energy efficiency, and fairness in multi-agent scenarios, showing great potential for improving communication efficiency and energy utilization.