It has always been difficult to achieve accurate information of the channel for underwater acoustic communications because of the severe underwater propagation conditions, including frequency-selective property, high relative mobility, long propagation latency, and intensive ambient noise, etc. To this end, a deep unfolding neural network based approach is proposed, in which multiple layers of the network mimic the iterations of the classical iterative sparse approximation algorithm to extract the inherent sparse features of the channel by exploiting deep learning, and a scheme based on the Sparsity-Aware DNN (SA-DNN) for UAC estimation is proposed to improve the estimation accuracy. Moreover, we propose a Denoising Sparsity-Aware DNN (DeSA-DNN) based enhanced method that integrates a denoising CNN module in the sparsity-aware deep network, so that the degradation brought by intensive ambient noise could be eliminated and the estimation accuracy can be further improved. Simulation results demonstrate that the performance of the proposed schemes is superior to the state-of-the-art compressed sensing based and iterative sparse recovery schems in the aspects of channel recovery precision, pilot overhead, and robustness, particularly under unideal circumstances of intensive ambient noise or inadequate measurement pilots.