In orthogonal multiple access (OMA) communication systems, resources are allocated to numerous users based on time, frequency, or code domains. The low bandwidth of the underwater acoustic network limits the number of nodes that can be supported by the OMA system, due to limited resources. The innovative concept of nonorthogonal multiple access (NOMA) provides a solution that offers more nodes and increases spectral efficiency. As a promising technique, it optimizes power allocation through channel characteristics and adopts serial interference cancellation algorithms to decode the signal at the receiver. This paper introduces an underwater communication system based on the node pairing algorithm of maximum and minimum weight (MMW-NOMA). First, we build the system model in a UAN scenario. Second, we present the node pairing strategy. Third, we design a node replacement mechanism. Furthermore, we offer a power allocation algorithm. Finally, we compare the performance of MMW-NOMA with randomly paired NOMA (RP-NOMA) and orthogonal frequency division multiplexing (OFDM). Numerical results show that MMW-NOMA outperforms RP-NOMA and OFDM in throughput, mean square error, and energy efficiency. However, the complexity of MMW-NOMA is slightly higher than that of RP-NOMA but significantly lower than OFDM. These results show that MMW-NOMA achieves a reasonable trade-off between performance and complexity.