The prediction as well as the gating of respiratory motion have received much attention over the last two decades for reducing the targeting error of the radiation treatment beam due to respiratory motion. In this article, we present a real-time algorithm for predicting respiratory motion in 3D space and realizing a gating function without pre-specifying a particular phase of the patient's breathing cycle. The algorithm, named EKF-GPRN(+) , first employs an extended Kalman filter (EKF) independently along each coordinate to predict the respiratory motion and then uses a Gaussian process regression network (GPRN) to correct the prediction error of the EKF in 3D space. The GPRN is a nonparametric Bayesian algorithm for modeling input-dependent correlations between the output variables in multi-output regression. Inference in GPRN is intractable and we employ variational inference with mean field approximation to compute an approximate predictive mean and predictive covariance matrix. The approximate predictive mean is used to correct the prediction error of the EKF. The trace of the approximate predictive covariance matrix is utilized to capture the uncertainty in EKF-GPRN(+) prediction error and systematically identify breathing points with a higher probability of large prediction error in advance. This identification enables us to pause the treatment beam over such instances. EKF-GPRN(+) implements a gating function by using simple calculations based on the trace of the predictive covariance matrix. Extensive numerical experiments are performed based on a large database of 304 respiratory motion traces to evaluate EKF-GPRN(+) . The experimental results show that the EKF-GPRN(+) algorithm reduces the patient-wise prediction error to 38%, 40% and 40% in root-mean-square, compared to no prediction, at lookahead lengths of 192 ms, 384 ms and 576 ms, respectively. The EKF-GPRN(+) algorithm can further reduce the prediction error by employing the gating function, albeit at the cost of reduced duty cycle. The error reduction allows the clinical target volume to planning target volume (CTV-PTV) margin to be reduced, leading to decreased normal-tissue toxicity and possible dose escalation. The CTV-PTV margin is also evaluated to quantify clinical benefits of EKF-GPRN(+) prediction.