In underwater acoustic (UWA) communication, orthogonal frequency division multiplexing (OFDM) is commonly employed to mitigate the inter-symbol interference (ISI) caused by delay spread. However, path-specific Doppler effects in UWA channels could result in significant inter-carrier interference (ICI) in the OFDM system. To address this problem, we introduce a multi-resolution convolutional neural network (CNN) named UWAModNet in this paper, designed to optimize the modem structure, specifically modulation and demodulation matrices. Based on a trade-off between the minimum and the average equivalent sub-channel rate, we propose an optimization criterion suitable to evaluate the performance of our learned modem. Additionally, a two-stage training strategy is developed to achieve quasi-optimal results. Simulations indicate that the learned modem outperforms zero-padded OFDM (ZP-OFDM) in terms of equivalent sub-channel rate and bit error rate, even under more severe Doppler effects during testing compared to training.