With the advancement of space missions and increasing complexity of spacecraft systems, traditional development methods that rely on experience and past examples are approaching the limits. A flexible and robust design space search method based on a systematic approach is required to accomplish challenging space missions. This paper presents a global trajectory optimization framework via a multi-fidelity approach that utilizes a graphics processing unit (GPU) for low-fidelity initial solution search and a central processing unit (CPU) to determine high-fidelity feasible solutions compliant with imposed constraints. A mission scenario employing transfer from a near-rectilinear halo orbit (NRHO) to a low lunar orbit (LLO) is considered to demonstrate the proposed framework, which consists of the following specific processes: (1) identifying a multitude of feasible trajectories as potential global optimum solutions with the aid of super-parallelized trajectory propagation using single-precision GPU cores; and then (2) determining accurate trajectories by means of gradient-based optimization incorporating double-precision propagation using CPU cores. The resultant trajectories are assessed via machine learning to identify the clustering structure, and verified in the light of the primer vector theory that evaluates local optimality in terms of minimum fuel consumption.