This study presents a recursive biomechanical model that validates that spiral phyllotaxis can be explained solely by Newtonian mechanics, without invoking additional forces or interactions. The model conceptualizes phyllotactic elements as expanding, non-deformable circles, driven by two primary forces: growth pressure, resulting from their continuous expansion, and outside pressure, arising from surrounding medium resistance. Results demonstrate that spiral structures emerge naturally, with patterns corresponding predominantly to Fibonacci and Lucas sequences. The computational implementation of the model demonstrates its capacity to produce biologically realistic patterns and reveals key insights into the dynamics of pattern formation, such as the stabilization of divergence angles and the emergence of densely packed structures. The study includes step-by-step video demonstrations and functional program code, allowing independent verification by researchers.