The blockchain paradigm provides a mechanism for content dissemination and distributed consensus on Peer-to-Peer (P2P) networks. While this paradigm has been widely adopted in industry, it has not been carefully analyzed in terms of its network scaling with respect to the number of peers. Applications for blockchain systems, such as cryptocurrencies and IoT, require this form of network scaling. In this paper, we propose a new stochastic network model for a blockchain system. We identify a structural property called \emph{one-endedness}, which we show to be desirable in any blockchain system as it is directly related to distributed consensus among the peers. We show that the stochastic stability of the network is sufficient for the one-endedness of a blockchain. We further establish that our model belongs to a class of network models, called monotone separable models. This allows us to establish upper and lower bounds on the stability region. The bounds on stability depend on the connectivity of the P2P network through its conductance and allow us to analyze the scalability of blockchain systems on large P2P networks. We verify our theoretical insights using both synthetic data and real data from the Bitcoin network.