We consider the Laplace-Beltrami operator $Δ_g$ on a smooth, compact Riemannian manifold $(M,g)$ and the determinantal point process $\mathcal{X}_λ$ on $M$ associated with the spectral projection of $-Δ_g$ onto the subspace corresponding to the eigenvalues up to $λ^2$. We show that the pull-back of $\mathcal{X}_λ$ by the exponential map $\exp_p : T_p^*M \to M$ under a suitable scaling converges weakly to the universal determinantal point process on $T_p^* M$ as $λ\to \infty$.