Weakly supervised temporal video grounding aims to localize query-relevant segments in untrimmed videos using only video-sentence pairs, without requiring ground-truth segment annotations that specify exact temporal boundaries. Recent approaches tackle this task by utilizing Gaussian-based temporal proposals to represent query-relevant segments. However, their inference strategies rely on heuristic mappings from Gaussian parameters to segment boundaries, resulting in suboptimal localization performance. To address this issue, we propose Gaussian Boundary Optimization (GBO), a novel inference framework that predicts segment boundaries by solving a principled optimization problem that balances proposal coverage and segment compactness. We derive a closed-form solution for this problem and rigorously analyze the optimality conditions under varying penalty regimes. Beyond its theoretical foundations, GBO offers several practical advantages: it is training-free and compatible with both single-Gaussian and mixture-based proposal architectures. Our experiments show that GBO significantly improves localization, achieving state-of-the-art results across standard benchmarks. Extensive experiments demonstrate the efficiency and generalizability of GBO across various proposal schemes. The code is available at https://github.com/sunoh-kim/gbo.