Calcium carbonate (CaCO₃) has attracted considerable attention owing to its structural versatility and broad applications in materials science and geochemistry. In this study, we employed Density Functional Theory (DFT) simulations to systematically investigate the structural, elastic, and dynamic properties of trigonal CaCO₃ under hydrostatic pressures ranging from 0 to 1.2 GPa. The optimized lattice constants closely align with previous theoretical and experimental values, thereby confirming the reliability of the computational approach. Mechanical stability was validated across the entire pressure range, with elastic constants and moduli demonstrating gradual increases under compressive strain. Elastic anisotropy was rigorously quantified using universal anisotropy indices, three-dimensional surface visualizations, and directional projections of elastic moduli. These analyses revealed pronounced pressure-dependent anisotropy. Furthermore, optical properties, including refractive indices and dielectric functions, were analyzed to clarify pressure-induced variations in electromagnetic interactions. These findings offer valuable insights into the pressure behavior of CaCO₃, advancing its potential applications in advanced functional materials and geophysical research.