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We analyze a secure two-hop mixed radio frequency (RF) and underwater wireless optical communication (UWOC) system using a fixed-gain amplify-and-forward (AF) relay. The UWOC channel is modeled using a unified mixture exponential-generalized Gamma distribution to consider the combined effects of air bubbles and temperature gradients on transmission characteristics. Both legitimate and eavesdropping RF channels are modeled using flexible $α-μ$ distributions. Specifically, we first derive both the probability density function (PDF) and cumulative distribution function (CDF) of the received signal-to-noise ratio (SNR) of the mixed RF and UWOC system. Based on the PDF and CDF expressions, we derive the closed-form expressions for the tight lower bound of the secrecy outage probability (SOP) and the probability of non-zero secrecy capacity (PNZ), which are both expressed in terms bivariate Fox's $H$-function. To utilize these analytical expressions, we derive asymptotic expressions of SOP and PNZ using only elementary functions. Also, we use asymptotic expressions to determine the optimal transmitting power to maximize energy efficiency. Further, we thoroughly investigate the effect of levels of air bubbles and temperature gradients in the UWOC channel, and study nonlinear characteristics of the transmission medium and the number of multipath clusters of the RF channel on the secrecy performance. Finally, all analyses are validated using Monte Carlo simulation.