The end of Dennard scaling and the slowdown of Moore's law led to a shift in technology trends toward parallel architectures, particularly in HPC systems. To continue providing performance benefits, HPC should embrace Approximate Computing (AC), which trades application quality loss for improved performance. However, existing AC techniques have not been extensively applied and evaluated in state-of-the-art hardware architectures such as GPUs, the primary execution vehicle for HPC applications today. This paper presents HPAC-Offload, a pragma-based programming model that extends OpenMP offload applications to support AC techniques, allowing portable approximations across different GPU architectures. We conduct a comprehensive performance analysis of HPAC-Offload across GPU-accelerated HPC applications, revealing that AC techniques can significantly accelerate HPC applications (1.64x LULESH on AMD, 1.57x NVIDIA) with minimal quality loss (0.1%). Our analysis offers deep insights into the performance of GPU-based AC that guide the future development of AC algorithms and systems for these architectures.