We propose a provably convergent algorithm for approximating the 2-Wasserstein barycenter of continuous non-parametric probability measures. Our algorithm is inspired by the fixed-point iterative scheme of Álvarez-Esteban et al. (2016) whose convergence to the 2-Wasserstein barycenter relies on obtaining exact optimal transport (OT) maps. However, typically in practice, OT maps are only approximately computed and exact computation of OT maps between continuous probability measures is only tractable for certain restrictive parametric families. To circumvent the need to compute exact OT maps between general non-parametric measures, we develop a tailored iterative scheme that utilizes consistent estimators of the OT maps instead of the exact OT maps. This gives rise to a computationally tractable stochastic fixed-point algorithm which is provably convergent to the 2-Wasserstein barycenter. Our algorithm remarkably does not restrict the support of the 2-Wasserstein barycenter to be any fixed finite set and can be implemented in a distributed computing environment, which makes it suitable for large-scale data aggregation problems. In our numerical experiments, we propose a method of generating non-trivial instances of 2-Wasserstein barycenter problems where the ground-truth barycenter measure is known. Our numerical results showcase the capability of our algorithm in developing high-quality approximations of the 2-Wasserstein barycenter, as well as its superiority over state-of-the-art methods based on generative neural networks in terms of accuracy, stability, and efficiency.