In this paper we consider a class of probability distributions on the six-vertex model from statistical mechanics, which originate from the higher spin vertex models of https://arxiv.org/abs/1601.05770. We define operators, inspired by the Macdonald difference operators, which extract various correlation functions, measuring the probability of observing different arrow configurations. The development of our operators is largely based on the properties of a remarkable family of symmetric rational functions, which were previously studied in https://arxiv.org/abs/1410.0976. For the class of models we consider, the correlation functions can be expressed in terms of multiple contour integrals, which are suitable for asymptotic analysis. For a particular choice of parameters we analyze the limit of the correlation functions through a steepest descent method. Combining this asymptotic statement with some new results about Gibbs measures on Gelfand-Tsetlin cones and patterns, we show that the asymptotic behavior of our six-vertex model near the boundary is described by the GUE-corners process.