Link prediction aims to reveal missing edges in a graph. We address this task with a deep graph convolutional Gaussian process model. The Gaussian process is transformed using simplified graph convolutions to better leverage the topological information of the graph domain. To scale the Gaussian process model to larger graphs, we introduce a variational inducing point method that places pseudo-inputs on a graph-structured domain. The proposed model represents the first Gaussian process for link prediction that can make use of both node features and topological information. We evaluate our model on three graph data sets with up to thousands of nodes and report consistent improvements over existing Gaussian process models and state-of-the-art graph neural network approaches.