Large language models (LLMs) are increasingly used in high-stakes domains, yet their unfaithful or non-factual outputs create serious risks and hinder trustworthy deployment. Rigorous fact-checking is therefore essential to mitigate these risks. Most existing factuality verification methods improve evidence acquisition, but fail to capture complex logical relations between claims and multi-source evidence, especially when different evidence sources are inconsistent or incomplete. Moreover, current benchmarks rarely capture such realistic conflict scenarios. To address this, we introduce MAVEN, a real-world benchmark for verifying complex claims against heterogeneous and conflicting web evidence with fine-grained explanations. We further propose ArgCheck, an argumentation-based fact-checking framework. It models claims and retrieved evidence as arguments linked by attack relations, resolving conflicts through formal argumentation semantics. To avoid incomplete evidence, ArgCheck incorporates adversarial web retrieval to gather balanced multi-source evidence and grounds its verdicts and explanations in accepted arguments. Experiments on MAVEN and the public fact-checking benchmark AVeriTeC show that ArgCheck improves Macro-F1 over the strongest baseline by 5.39 and 11.10 points, respectively, while generating more faithful and evidence-grounded explanations. Our code and dataset are available online https://github.com/ZyraChen/ArgCheck.