Social media platforms have become critical arenas for public discourse, yet existing stance detection methods often reduce opinions to surface-level labels, overlooking the conversational evidence behind stance expressions. We introduce Conversational Stance-Cause Pair Detection (CSCPD), a new task that jointly identifies both the stance polarity and its observable contextual evidence within multi-turn conversations. To advance research in this direction, we present Cause-CSD, the first large-scale dataset for CSCPD, spanning 21,048 annotated stance-cause pairs across diverse open-domain, textual, and multimodal discussions. We further propose Stance-Cause Detection Language Model (SCD-LM), a unified language model framework that leverages explicit context reasoning and joint decoding to predict stances and their supporting causes, along with human-readable rationales. Extensive experiments demonstrate that SCD-LM achieves state-of-the-art results on both text-only and multimodal subtasks, significantly outperforming strong baselines, especially for long-range and image-grounded cause detection. Our work advances explainable stance analysis and underpins understanding of public opinion drivers in impactful online settings.