The interplay between cellular mechanics and other biochemical processes leading to the cell cycle is poorly understood. Here, we propose a quantitative model of the mechanism of cell budding, particularly for Saccharomyces cerevisiae, as a ''weaken-fill-repair'' process. This model links Newtonian mechanics over the cell wall with biochemical processes that change the mechanical properties of the wall. Our model shows that: (1) The oscillatory behavior in the size of the mother compartment during cell division, as observed experimentally, is an inevitable consequence of the budding mechanism; (2) The asymmetry of cell division is necessary for the daughter cell to maintain mechanical stiffness; (3) Although different aspects of a cell are strictly constrained and intricately interconnected, the budding process is governed by a single reduced parameter that, when at an appropriate value, can automatically achieve homeostasis. Our quantitative model of the evolution of cell walls and their impact on cell division could provide valuable insights into cell morphology from a system perspective. This knowledge could be instrumental in developing effective antimicrobial treatments against microorganisms with cell walls and even in building synthetic cells.