Differential privacy has become a cornerstone in the development of privacy-preserving learning algorithms. This work addresses optimizing differentially private kernel learning within the empirical risk minimization (ERM) framework. We propose a novel differentially private kernel ERM algorithm based on random projection in the reproducing kernel Hilbert space using Gaussian processes. Our method achieves minimax-optimal excess risk rates for both the squared loss and Lipschitz-smooth convex loss functions under a local strong convexity condition. We further show that existing approaches based on alternative dimension reduction techniques, such as random Fourier feature mappings or $\ell_2$ regularization, yield suboptimal excess risk bounds. Our key theoretical contribution also includes the derivation of dimension-free excess risk bounds for objective perturbation-based private linear ERM, marking the first such result that does not rely on noisy gradient-based mechanisms. Additionally, we obtain sharper excess risk bounds for existing differentially private kernel ERM algorithms. Empirical evaluations support our theoretical claims, demonstrating that random projection enables statistically efficient and optimally private kernel learning. These findings provide new insights into the design of differentially private algorithms and highlight the central role of dimension reduction in balancing privacy and utility.