We investigate the origin of causal structure and the finite speed of signal propagation within the framework of Universal Modular Dynamics (UMD). Starting from the density operator ρ as the fundamental carrier of physical information, we analyze the dynamics of correlations under modular evolution governed by a CPTPconsistent equation. We show that constraints on the rate of correlation transport impose a fundamental bound on propagation speed. This bound defines an emergent causal speed, which can be identified with the speed of light c, not as a postulate but as a derived quantity. The resulting structure induces an effective causal cone analogous to the light cone of relativistic physics, establishing causality as an emergent property of informational dynamics. This provides a derivation of causal structure from first principles and challenges the conventional view that causality is fundamental, within the domain of validity of the framework. We further demonstrate that consistency between entropy growth, correlation flow, and causal structure constrains the evolution of emergent geometry. These constraints provide a pathway toward the derivation of Einstein dynamics from purely informational principles. Our results suggest that both causality and gravitational dynamics arise from fundamental limits on information propagation, establishing a direct bridge between quantum information, geometry, and gravity.