The advancement of underwater cognitive acoustic network (UCAN) technology aims to improve spectral efficiency and ensure coexistence with the underwater ecosystem. As the demand for short-term underwater applications operated under distributed topologies, like autonomous underwater vehicle cluster operations, continues to grow, this paper presents Underwater Multi-channel Medium Access Control with Cognitive Acoustics (UMMAC-CA) as a suitable channel access protocol for distributed UCANs. UMMAC-CA operates on a per-frame basis, similar to the Multi-channel Medium Access Control with Cognitive Radios (MMAC-CR) designed for distributed cognitive radio networks, but with notable differences. It employs a pre-determined data transmission matrix to allow all nodes to access the channel without contention, thus reducing the channel access overhead. In addition, to mitigate the communication failures caused by randomly occurring interferers, UMMAC-CA allocates at least 50% of frame time for interferer sensing. This is possible because of the fixed data transmission scheduling, which allows other nodes to sense for interferers simultaneously while a specific node is transmitting data. Simulation results demonstrate that UMMAC-CA outperforms MMAC-CR across various metrics, including those of the sensing time rate, controlling time rate, and throughput. In addition, except for in the case where the data transmission time coefficient equals 1, the message overhead performance of UMMAC-CA is also superior to that of MMAC-CR. These results underscore the suitability of UMMAC-CA for use in challenging underwater applications requiring multi-channel cognitive communication within a distributed network architecture.