To bridge the digital divide, space-ground integrated networks (SGINs) are expected to deliver artificial intelligence (AI) services to every corner of the world. One key mission of SGINs is to support federated learning (FL) at a global scale. However, existing space-ground integrated FL frameworks involve ground stations or costly inter-satellite links, entailing excessive training latency and communication costs. To overcome these limitations, we propose an infrastructure-free federated learning framework based on a model dispersal (FedMeld) strategy, which exploits periodic movement patterns and store-carry-forward capabilities of satellites to enable parameter mixing across large-scale geographical regions. We theoretically show that FedMeld leads to global model convergence and quantify the effects of round interval and mixing ratio between adjacent areas on its learning performance. Based on the theoretical results, we formulate a joint optimization problem to design the staleness control and mixing ratio (SC-MR) for minimizing the training loss. By decomposing the problem into sequential SC and MR subproblems without compromising the optimality, we derive the round interval solution in a closed form and the mixing ratio in a semi-closed form to achieve the optimal latency-accuracy tradeoff. Experiments using various datasets demonstrate that FedMeld achieves superior model accuracy while significantly reducing communication costs as compared with traditional FL schemes for SGINs.