Actin-based motility is widespread among eukaryotic cells. Many distinct cell types must undergo migration reliably in order to perform a variety of biological functions. However, the fundamental mechanism by which actin produces force, the Brownian ratchet, carries with it an inherent stochasticity. We aimed to understand how robust migration behavior can arise from stochastic actin polymerization in the context of lamellipodial motility. In previous experimental and theoretical efforts, we developed a stochastic model of the leading edge that faithfully recapitulates the stable lamellipodial shape dynamics observed in migrating cells. The stability of this minimal model, in the absence of additional feedback mechanisms, necessarily implies that lamellipodial maintenance is an intrinsic property of branched actin growth against a membrane. In the present work, we performed a theoretical exploration into the biophysical mechanisms underlying stability in our model. Surprisingly, we find that membrane tension is dispensable for maintaining a stable leading edge, and instead identify lateral filament spreading, mediated by the inherent geometry of branched actin growth, as the critical contributor to lamellipodial stability. Furthermore, we find that the genetically encoded branch geometry (the Arp2/3-mediated 70-80° branching angle, conserved from protists to mammals) maximally suppresses leading edge shape and actin density fluctuations. Taken together, these results contextualize two long-standing experimental observations: First, they suggest that the essential lamellipodial maintenance role of the Arp2/3 complex is rooted in its ability to facilitate filament spreading and suppress stochastic fluctuations. Second, the results hint that the high conservation of the Arp2/3-mediated branching angle may stem from its capacity to optimally perform this smoothing effect. In conclusion, we find that the branched structure of actin at the leading edge naturally quells stochastic fluctuations, thereby representing a novel biological noise-suppression mechanism based entirely on system geometry.