VC dimension for Gaussian Process Regression
The expressiveness of the gaussian process grows with a number of training points. So, the vapnik-chervonenkis dimension in fact is infinite (pretty much the same way it's infinite for k nearest neighbors) and unfortunately your rule of thumb is not applicable here.
You should probably rely on train/validation split to estimate the generalization. From my experience, GP generalizes much better than neural nets, but the exact required train size depends on data distribution complexity itself.