The silica-based cell walls of diatoms are prime examples of genetically controlled, species-specific mineral architectures. The physical principles underlying morphogenesis of their hierarchically structured silica patterns are not understood, yet such insight could reveal novel routes towards synthesizing functional inorganic materials. Recent advances in imaging nascent diatom silica allow rationalizing possible mechanisms of their pattern formation. Here, we combine theory and experiments on the model diatom Thalassiosira pseudonana to put forward a minimal model for morphogenesis of branched rib patterns – a widespread feature of diatom cell walls. To this end, we developed an automated image analysis algorithm that enabled quantitative assessment of the morphological discrepancy between the experiments and model predictions. The model proposed here quantitatively recapitulates the time-course of rib pattern formation by accounting for silica biochemistry with autocatalytic formation of diffusible silica precursors followed by conversion into solid silica. We propose that silica deposition releases an inhibitor that slows down up-stream precursor conversion, thereby implementing a self-replicating reaction-diffusion system, recapitulated by a non-classical Turing mechanism. The proposed mechanism highlights the role of geometrical cues for guided self-organization, rationalizing the instructive role for the single initial pattern seed known as primary silicification site present in diatoms. The model features a wide spectrum of possible pattern morphologies depending on the model parameters, suggesting that this model may be applicable in other diatom species. Moreover, due to the generic nature of the proposed model for branching morphogenesis, the mechanism identified here may be relevant also in other biological systems known to exhibit.