Affine registration is indispensable in a comprehensive medical image registration pipeline. However, only a few studies focus on fast and robust affine registration algorithms. Most of these studies utilize convolutional neural networks (CNNs) to learn joint affine and non-parametric registration, while the standalone performance of the affine subnetwork is less explored. Moreover, existing CNN-based affine registration approaches focus either on the local misalignment or the global orientation and position of the input to predict the affine transformation matrix, which are sensitive to spatial initialization and exhibit limited generalizability apart from the training dataset. In this paper, we present a fast and robust learning-based algorithm, Coarse-to-Fine Vision Transformer (C2FViT), for 3D affine medical image registration. Our method naturally leverages the global connectivity and locality of the convolutional vision transformer and the multi-resolution strategy to learn the global affine registration. We evaluate our method on 3D brain atlas registration and template-matching normalization. Comprehensive results demonstrate that our method is superior to the existing CNNs-based affine registration methods in terms of registration accuracy, robustness and generalizability while preserving the runtime advantage of the learning-based methods. The source code is available at this https URL.