Rumor detection on social media remains a challenging task due to the complex propagation dynamics and the limited interpretability of existing models. While recent neural architectures capture content and structural features, they often fail to reveal the underlying causal mechanisms of misinformation spread. We propose CausalMamba, a novel framework that integrates Mamba-based sequence modeling, graph convolutional networks (GCNs), and differentiable causal discovery via NOTEARS. CausalMamba learns joint representations of temporal tweet sequences and reply structures, while uncovering latent causal graphs to identify influential nodes within each propagation chain. Experiments on the Twitter15 dataset show that our model achieves competitive classification performance compared to strong baselines, and uniquely enables counterfactual intervention analysis. Qualitative results demonstrate that removing top-ranked causal nodes significantly alters graph connectivity, offering interpretable insights into rumor dynamics. Our framework provides a unified approach for rumor classification and influence analysis, paving the way for more explainable and actionable misinformation detection systems.