This work pioneers a comprehensive performance analysis of direct air-water optical wireless communication (OWC) systems employing vertically stratified underwater channels. Specifically, the atmospheric turbulence of FSO link is modified by Fisher-Snedecor F distribution, and the air-water interface is assumed to follow Birnbaum-Saunders (BS) distribution, meanwhile the oceanic turbulence is described by cased Gamma-Gamma distribution. The path loss, geometric spread and pointing errors between the transmitter and receiver are also taken into account. Considering both perfect and imperfect channel state information (CSI) conditions, novel closed-form expressions for outage probability (OP), average bit error rate (ABER), and average capacity are derived, utilizing bivariate Meijer G- and Fox""s H-functions. Moreover, the asymptotic analyses for the OP and ABER are presented to gain more insights. Monte Carlo simulations are also provided to confirm the obtained analytical results. This work will benefit the research in the field of the air-to-water OWC.