Traditional methods used in causal mediation analysis with continuous treatment often focus on estimating average causal effects, limiting their applicability in precision medicine. Machine learning techniques have emerged as a powerful approach for precisely estimating individualized causal effects. This paper proposes a novel method called CGAN-ICMA-CT that leverages Conditional Generative Adversarial Networks (CGANs) to infer individualized causal effects with continuous treatment. We thoroughly investigate the convergence properties of CGAN-ICMA-CT and show that the estimated distribution of our inferential conditional generator converges to the true conditional distribution under mild conditions. We conduct numerical experiments to validate the effectiveness of CGAN-ICMA-CT and compare it with four commonly used methods: linear regression, support vector machine regression, decision tree, and random forest regression. The results demonstrate that CGAN-ICMA-CT outperforms these methods regarding accuracy and precision. Furthermore, we apply the CGAN-ICMA-CT model to the real-world Job Corps dataset, showcasing its practical utility. By utilizing CGAN-ICMA-CT, we estimate the individualized causal effects of the Job Corps program on the number of arrests, providing insights into both direct effects and effects mediated through intermediate variables. Our findings confirm the potential of CGAN-ICMA-CT in advancing individualized causal mediation analysis with continuous treatment in precision medicine settings.