Topology optimization for stress minimization of large-scale 3D structures remains highly challenging and has rarely been explored. The primary difficulties involve the need for high-resolution discretization to accurately resolve stress distributions and the algorithmic capability to eliminate localized stress concentrations. We propose a GPU-accelerated floating projection topology optimization (FPTO) framework to tackle large-scale 3D stress minimization problems using linear material interpolation. The framework leverages a combination of Matlab and CUDA C kernels, where performance-critical tasks, including finite element analysis, stress analysis, and sensitivity analysis, are offloaded to the GPU to maximize computational efficiency. To reduce memory usage and accelerate calculations, a matrix-free multigrid preconditioned conjugate gradient (MGPCG) solver based on a linear homogenization technique is introduced. Numerical examples demonstrate the effectiveness and efficiency of the GPU-accelerated FPTO method for stress minimization of large-scale 3D structures. Compared with compliance-minimization designs, stress-minimization designs exhibit different topologies that not only effectively eliminate stress concentrations but also significantly reduce the maximum von Mises stress, thereby improving their strength. The proposed GPU-accelerated algorithm provides a solid foundation for the practical application of topology optimization.