Performing a high-capacity Medium Access Control (MAC) protocol suffers from low bandwidth and long propagation delay in Underwater Acoustic Networks (UANs). Non-Orthogonal Multiple Access (NOMA) is a promising technology to assist MAC protocols in overcoming the above restrictions and improving UANs’ capacity. It enables multiple users to access the same frequency-time resource based on power or code differences of user classes. However, UANs lack MAC research employing NOMA’s physical advantages. Most existing NOMA-based MAC protocols are designed for terrestrial networks, which are inapplicable to UANs. In classifying users, they ignore the effects of harsh marine environments on acoustic channels and fail to decrease channels’ interference, resulting in conflicting communications. Moreover, such unreasonable classification results further affect resource allocation, leading to low transmission rate and high energy consumption in UANs. In this paper, we propose a Hybrid NOMA-based MAC protocol (HN-MAC) to achieve efficient concurrent communication for UANs. Specifically, HN-MAC combines power-domain and code-domain NOMA to classify users and allocate communication resources. For the user classification, we propose an Adaptive Clustering Algorithm (ACA), which dynamically determines the clusters’ number and classifies users based on channel gain and channel correlation under multipath conditions. In this way, HN-MAC decreases interference among multiple users in various ocean scenarios. During the resource allocation, we formulate a joint allocation problem of transmission power and codebook based on the clustering result to optimize transmission rate and energy consumption. Further, a genetic algorithm is proposed to solve the allocation problem by considering resource constraints. Simulation results show that HN-MAC provides more stable concurrent communications with less resource consumption than the state-of-the-art protocols in various UANs.