A major open question in statistical mechanics, known as the Gaussian spin wave conjecture, predicts that the low temperature phase of the Abelian spin systems with continuous symmetry behave like Gaussian free fields. In this paper we consider the classical Villain rotator model in $\mathbb{Z}^d$, $d \ge 3$ at sufficiently low temperature, and prove that the truncated two-point function decays asymptotically as $|x|^{2−d}$, with an algebraic rate of convergence.We also obtain the same asymptotic decay separately for the transversal two-point functions. This quantifies the spontaneous magnetization result for the Villain model at low temperatures and constitutes a first step toward a more precise understanding of the spin-wave conjecture. We believe that our method extends to finite range interactions, and to other Abelian spin systems and Abelian gauge theory in $d \ge 3$. We also develop a quantitative perspective on homogenization of uniformly convex gradient Gibbs measures.