The unitary group with the Haar probability measure is called Circular Unitary Ensemble. All the eigenvalues lie on the unit circle in the complex plane and they can be regarded as a determinantal point process on $\mathbb{S}^1$. It is also known that the scaled point processes converge weakly to the determinantal point process associated with the so-called sine kernel as the size of matrices tends to $\infty$. We extend this result to the case of high-dimensional spheres and show that the scaling limit processes are determinantal point processes associated with the kernels expressed by the Bessel functions of the first kind.