We target at the task of weakly-supervised video object grounding (WSVOG), where only video-sentence annotations are available during model learning. It aims to localize objects described in the sentence to visual regions in the video, which is a fundamental capability needed in pattern analysis and machine learning. Despite the recent progress, existing methods all suffer from the severe problem of spurious association, which will harm the grounding performance. In this paper, we start from the definition of WSVOG and pinpoint the spurious association from two aspects: (1) the association itself is not object-relevant but extremely ambiguous due to weak supervision; and (2) the association is unavoidably confounded by the observational bias when taking the statistics-based matching strategy in existing methods. With this in mind, we design a unified causal framework to learn the deconfounded object-relevant association for more accurate and robust video object grounding. Specifically, we learn the object-relevant association by causal intervention from the perspective of video data generation process. To overcome the problems of lacking fine-grained supervision in terms of intervention, we propose a novel spatial-temporal adversarial contrastive learning paradigm. To further remove the accompanying confounding effect within the object-relevant association, we pursue the true causality by conducting causal intervention via backdoor adjustment. Finally, the deconfounded object-relevant association is learned and optimized under a unified causal framework in an end-to-end manner. Extensive experiments on both IID and OOD testing sets of three benchmarks demonstrate its accurate and robust grounding performance against state-of-the-arts.