This paper presents a novel human-robot interaction (HRI) framework that enables intuitive gesture-driven control through a capacitance-based woven tactile skin. Unlike conventional interfaces that rely on panels or handheld devices, the woven tactile skin integrates seamlessly with curved robot surfaces, enabling embodied interaction and narrowing the gap between human intent and robot response. Its woven design combines fabric-like flexibility with structural stability and dense multi-channel sensing through the interlaced conductive threads. Building on this capability, we define a gesture-action mapping of 14 single- and multi-touch gestures that cover representative robot commands, including task-space motion and auxiliary functions. A lightweight convolution-transformer model designed for gesture recognition in real time achieves an accuracy of near-100%, outperforming prior baseline approaches. Experiments on robot arm tasks, including pick-and-place and pouring, demonstrate that our system reduces task completion time by up to 57% compared with keyboard panels and teach pendants. Overall, our proposed framework demonstrates a practical pathway toward more natural and efficient embodied HRI.