In this paper, we extend and investigate the properties of the semi-smooth Newton method when applied to a general projection equation in finite dimensional spaces. We first present results concerning Clarke's generalized Jacobian of the projection onto a closed and convex cone. We then describe the iterative process for the general cone case and establish two convergence theorems. We apply these results to the constrained quadratic conic programming problem, emphasizing its connection to the projection equation. To illustrate the performance of our method, we conduct numerical experiments focusing on semidefinite least squares, in particular the nearest correlation matrix problem. In the latter scenario, we benchmark our outcomes against previous literature, presenting performance profiles and tabulated results for clarity and comparison.