The lightweight design of thin-walled curved structures as spherical shells are frequently implemented in architecture, aerospace, mechanical, automotive, nuclear, and defense structures. Under the transverse loads, shells may largely deform and hence snap from one equilibrium position to the other. Thus, this work aims to develop a mathematical model and computational solution to investigate the nonlinear bending and snap-through behavior of doubly curved auxetic metamaterial shell subjected to transversal loading, for the first time. The shell is composed of several layers through the shell thickness, each layer is manufactured from a copper (Cu) matrix reinforced with a specified weight fraction of graphene origami auxetic metamaterial (GOAM). The mechanical properties of the GOAM shell panel are presented and described by functions of the GOAM volume fraction and folding degree. Three types of GOAM-distributions are considered, which are U-type, X-type, and O-type. The theoretical framework of Kirchhoff–Love hypotheses for thin shells and von Karman type nonlinearity are used to derive the governing equations. The differential quadrature method (DQM) is implemented to discretize the space domain and convert the nonlinear partial differential equation to nonlinear algebraic equations in terms of displacement field. An efficient incremental iterative procedure is developed to solve the nonlinear equations and predict the snap-through behavior. A model conversion and validation with isotropic spherical shell is considered. Several numerical results are conducted considering the effects of changing GOAM content, distribution pattern, folding degree, and shell curvature and thickness. For panels exhibiting snap-through behavior, increasing the shell curvature leads to higher snap-through limiting load; however, the instability gap is enlarged.