Communication over acoustic signals underwater results in multi-scale multi-lag channels due to multipath propagation. Hence, a robust channel estimation technique has to be present at the receiver. In this paper, assuming underwater channels undergoing Rayleigh fading, a path-based channel model that characterizes each path of the time-varying sparse channel by a delay, a Doppler scale, and an attenuation factor is considered. Alamouti's space-time block transmit diversity scheme is used in the form of two transmit antennas and one receiver, and the proposed OFDM-based non-data-aided algorithm iteratively estimates the complex channel parameters of each subcarrier using the expectation maximization (EM) method, which in turn converges to a true maximum a posteriori probability (MAP) estimate of the unknown channel, where the Karhunen-Lo'eve expansion is performed for complexity reduction. Finally, the novel channel estimation algorithm combines the aforementioned MAP-EM technique with ESPRIT for delay estimation by exploiting the sparseness of the underwater acoustic channels. The performance of the proposed algorithm is then presented in terms of average mean square error and symbol error rate for QPSK signaling with extreme Doppler spreads and different pilot spacings. It is shown that excellent mean-square error and symbol error rate performance is achieved even in the presence of extreme Doppler shifts.