Model Predictive Control (MPC) is an advanced control method that is gaining interest thanks to its ability to control multivariable systems and handle constraints. Nevertheless, the effectiveness of MPC depends on the accuracy of the mathematical model that is used to make predictions about the system state trajectory. As the design of an accurate first-principle model of the plant to be controlled can be challenging, an increasing number of predictive controllers based on black-box models have been proposed. Among the black-box models that have been considered, Gaussian Processes (GPs) have been of particular relevance. However, the implications of GP model uncertainties on the recursive feasibility of predictive controllers and the stability of the closed-loop system are still a debated issue. In this manuscript, we resort to recent results about the stabilizing properties of robust MPC algorithms based on the contractivity of the system to be controlled, to build a robust MPC based on GPs that can guarantee stability and recursive feasibility under mild assumptions.