Gaussian processes (GPs) are widely employed as versatile modeling and predictive tools in spatial statistics, functional data analysis, computer modeling and diverse applications of machine learning. They have been widely studied over Euclidean spaces, where they are specified using covariance functions or covariograms for modelling complex dependencies. There is a growing literature on GPs over Riemannian manifolds in order to develop richer and more flexible inferential frameworks. While GPs have been extensively studied for asymptotic inference on Euclidean spaces using positive definite covariograms, such results are relatively sparse on Riemannian manifolds. We undertake analogous developments for GPs constructed over compact Riemannian manifolds. Building upon the recently introduced Matérn covariograms on a compact Riemannian manifold, we employ formal notions and conditions for the equivalence of two Matérn Gaussian random measures on compact manifolds to derive the microergodic parameters and formally establish the consistency of their maximum likelihood estimates as well as asymptotic optimality of the best linear unbiased predictor.