White matter fiber clustering is an important strategy for white matter parcellation, which enables quantitative analysis of brain connections in health and disease. In combination with expert neuroanatomical labeling, data-driven white matter fiber clustering is a powerful tool for creating atlases that can model white matter anatomy across individuals. While widely used fiber clustering approaches have shown good performance using classical unsupervised machine learning techniques, recent advances in deep learning reveal a promising direction toward fast and effective fiber clustering. In this work, we propose a novel deep learning framework for white matter fiber clustering, Deep Fiber Clustering (DFC), which solves the unsupervised clustering problem as a self-supervised learning task with a domain-specific pretext task to predict pairwise fiber distances. This process learns a high-dimensional embedding feature representation for each fiber, regardless of the order of fiber points reconstructed during tractography. We design a novel network architecture that represents input fibers as point clouds and allows the incorporation of additional sources of input information from gray matter parcellation. Thus, DFC makes use of combined information about white matter fiber geometry and gray matter anatomy to improve the anatomical coherence of fiber clusters. In addition, DFC conducts outlier removal naturally by rejecting fibers with low cluster assignment probability. We evaluate DFC on three independently acquired cohorts, including data from 220 individuals across genders, ages (young and elderly adults), and different health conditions (healthy control and multiple neuropsychiatric disorders). We compare DFC to several state-of-the-art white matter fiber clustering algorithms. Experimental results demonstrate superior performance of DFC in terms of cluster compactness, generalization ability, anatomical coherence, and computational efficiency.