A machine learning (ML) approach has been recently proposed to improve orbit prediction accuracy of resident space objects through learning from historical data. Previous results have shown that the ML approach can successfully improve the point estimation accuracy. This paper extends the ML approach by introducing the Gaussian process regression (GPR) method to generate uncertainty information about the point estimate. Numerical results demonstrate that GPR can effectively improve the orbit prediction accuracy and the generated uncertainty boundaries can cover the majority of the testing data. Additionally, effects of the number of the basis function used by the GPR and the orbital measurement noise are explored. Results reveal that properly designed and trained GPRs have stable performance for all experiment cases.