Underwater wireless optical communication (UWOC) is a potential technology for high-speed and reliable underwater data transmission. In underwater environment, ocean turbulence has a strong impact on the performance of UWOC systems. Thus, transmission laser selection (TLS) is proposed as an effective technique for fading mitigation in turbulence channels. In this paper, we present a vertical UWOC system with generalized TLS (GTLS) in multi-layer cascaded Gamma-Gamma ocean turbulence channels. With GTLS, the transmitter is equipped with N laser sources and the nth source is selected for transmission. For the ideal case, the optimal laser source is selected, but in practice, a sub-optimal or worse source may be selected due to channel estimation and/or feedback errors. For the GTLS system, we derive an exact closed-form expression for outage probability. Furthermore, based on the outage probability expression, the diversity order and asymptotical diversity order expression are analyzed. Finally, we give simulation results to validate our analytical results. Numerical results show that the GTLS system performs better than the system without GTLS (i.e. N = 1). The number of cascade channel layers, the number of laser sources and the selection of source index significantly affect the performance of the GTLS system.