Current spherical surface registration methods achieve good performance on alignment and spatial normalization of cortical surfaces across individuals in neuroimaging analysis. However, they are computationally intensive, since they have to optimize an objective function independently for each pair of surfaces. In this paper, we present a fast learning-based algorithm that makes use of the recent development in spherical Convolutional Neural Networks (CNNs) for spherical cortical surface registration. Given a set of surface pairs without supervised information such as ground truth deformation fields or anatomical landmarks, we formulate the registration as a parametric function and learn its parameters by enforcing the feature similarity between one surface and the other one warped by the estimated deformation field using the function. Then, given a new pair of surfaces, we can quickly infer the spherical deformation field registering one surface to the other one. We model this parametric function using three orthogonal Spherical U-Nets and use spherical transform layers to warp the spherical surfaces, while imposing smoothness constraints on the deformation field. All the layers in the network are well-defined and differentiable, thus the parameters can be effectively learned. We show that our method achieves accurate cortical alignment results on 102 subjects, comparable to two state-of-the-art methods: Spherical Demons and MSM, while runs much faster.