We present GoldbachGPU, an open-source framework for large-scale computational verification of Goldbach's conjecture using commodity GPU hardware. Prior GPU-based approaches reported a hard memory ceiling near 10^11 due to monolithic prime-table allocation. We show that this limitation is architectural rather than fundamental: a dense bit-packed prime representation provides a 16x reduction in memory footprint, and a segmented double-sieve design removes the VRAM ceiling entirely. By inverting the verification loop and combining a GPU fast-path with a multi-phase primality oracle, the framework achieves exhaustive verification up to 10^12 on a single NVIDIA RTX 3070 (8 GB VRAM), with no counterexamples found. Each segment requires 14 MB of VRAM, yielding O(N) wall-clock time and O(1) memory in N. A rigorous CPU fallback guarantees mathematical completeness, though it was never invoked in practice. An arbitrary-precision checker using GMP and OpenMP extends single-number verification to 10^10000 via a synchronised batch-search strategy. The segmented architecture also exhibits clean multi-GPU scaling on data-centre hardware (tested on 8 x H100). All code is open-source, documented, and reproducible on both commodity and high-end hardware.