Deep learning-based fringe projection profilometry (FPP) shows potential for challenging three-dimensional (3D) reconstruction of objects with dynamic motion, complex surface, and extreme environment. However, the previous deep learning-based methods are all supervised ones, which are difficult to be applied for scenes that are different from the training, thus requiring a large number of training datasets. In this paper, we propose a new geometric constraint-based phase unwrapping (GCPU) method that enables an untrained deep learning-based FPP for the first time. An untrained convolutional neural network is designed to achieve correct phase unwrapping through a network parameter space optimization. The loss function of the optimization is constructed by following the 3D, structural, and phase consistency. The designed untrained network directly outputs the desired fringe order with the inputted phase and fringe background. The experiments verify that the proposed GCPU method provides higher robustness compared with the traditional GCPU methods, thus resulting in accurate 3D reconstruction for objects with a complex surface. Unlike the commonly used temporal phase unwrapping, the proposed GCPU method does not require additional fringe patterns, which can also be used for the dynamic 3D measurement.