The novel technique of distributed acoustic sensing (DAS) holds great potential for underwater seismology by transforming standard telecommunication cables, such as those currently traversing most of the world's oceans, into dense arrays of seismo-acoustic sensors. To harness these measurements for seismic monitoring, the ability to record transient ground deformations using telecommunication fibers is investigated here by analyzing ambient noise, earthquake signals, and their associated phase velocities, on DAS records from three dark fibers in the Mediterranean Sea. The recording quality varies dramatically along the fibers and is strongly correlated with the bathymetry and the apparent phase velocities of the recorded waves. Apparent velocities are determined for several well-recorded earthquakes and used to convert DAS S-wave strain spectra to ground motion spectra. Excellent agreement is found between the spectra of nearby underwater and on-land seismometers and DAS converted spectra, when the latter are corrected for site effects. Apparent velocities greatly affect the ability to detect seismic deformations: for the same ground motions, slower waves induce higher strains and thus are more favorably detected than fast waves. The effect of apparent velocity on the ability to detect seismic phases, quantified by expected signal-to-noise ratios, is investigated by comparing signal amplitudes predicted by an earthquake ground motion model to recorded noise levels. DAS detection capabilities on underwater fibers are found to be similar to those of nearby broadband sensors, and superior to those of on-land fiber segments. The results demonstrate the great potential of underwater DAS for seismic monitoring and earthquake early warning.