Manual classification of functional resting state networks (RSNs) derived from Independent Component Analysis (ICA) decomposition can be labor intensive and requires expertise, particularly in large multi-subject analyses. Hence, a fully automatic algorithm that can reliably classify these RSNs is desirable. In this paper, we present a deep learning approach based on a Siamese Network to learn a discriminative feature representation for single-subject ICA component classification. Advantages of this supervised framework are that it requires relatively few training data examples and it does not require the number of ICA components to be specified. In addition, our approach permits one-shot learning, which allows generalization to new classes not seen in the training set with only one example of each new class. The proposed method is shown to out-perform traditional convolutional neural network (CNN) and template matching methods in identifying eleven subject-specific RSNs, achieving 100% accuracy on a holdout data set and over 99% accuracy on an outside data set. We also demonstrate that the method is robust to scan-rescan variation. Finally, we show that the functional connectivity of default mode and salience networks identified by the proposed technique is altered in a group analysis of mild traumatic brain injury (TBI), severe TBI, and healthy subjects.