Distributed massive multiple-input multiple output (mMIMO) system for low earth orbit (LEO) satellite networks is introduced as a promising technique to provide broadband connectivity. Nevertheless, several challenges persist in implementing distributed mMIMO systems for LEO satellite networks. These challenges include providing scalable massive access implementation as the system complexity increases with network size. Another challenging issue is the asynchronous arrival of signals at the user terminals due to the different propagation delays among distributed antennas in space, which destroys the coherent transmission, and consequently degrades the system performance. In this paper, we propose a scalable distributed mMIMO system for LEO satellite networks based on dynamic user-centric clustering. Aiming to obtain scalable implementation, new algorithms for initial cooperative access, cluster selection, and cluster handover are provided. In addition, phase shift-aware precoding is implemented to compensate for the propagation delay phase shifts. The performance of the proposed user-centric distributed mMIMO is compared with two baseline configurations: the non-cooperative transmission systems, where each user connects to only a single satellite, and the full-cooperative distributed mMIMO systems, where all satellites contribute serving each user. The numerical results show the potential of the proposed distributed mMIMO system to enhance system spectral efficiency when compared to noncooperative transmission systems. Additionally, it demonstrates the ability to minimize the serving cluster size for each user, thereby reducing the overall system complexity in comparison to the full-cooperative distributed mMIMO systems.