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In a channel shared by several nodes, the scheduling algorithm is a key factor to avoiding collisions in the random access-based approach. Commonly, scheduling algorithms can be used to enhance network performance to meet certain requirements. Therefore, in this paper we propose a Delay-Aware Media Access Control (DAMAC) protocol for monitoring time-sensitive applications over multi-hop in Underwater Acoustic Sensor Networks (UASNs), which relies on the random access-based approach where each node uses Carrier Sense Multiple Access/Collision Avoidance (CSMA/CA) to determine channel status, switches nodes on and off to conserve energy, and allows concurrent transmissions to improve the underwater communication in the UASNs. In addition, DAMAC does not require any handshaking packets prior to data transmission, which helps to improve network performance in several metrics. The proposed protocol considers the long propagation delay to allow concurrent transmissions, meaning nodes are scheduled to transmit their data packets concurrently to exploit the long propagation delay between underwater nodes. The simulation results show that DAMAC protocol outperforms Aloha, BroadcastMAC, RMAC, Tu-MAC, and OPMAC protocols under varying network loads in terms of energy efficiency, communication overhead, and fairness of the network by up to 65%, 45%, and 726%, respectively.