The Schrodinger-Newton system is a nonlinear system obtained by coupling together the linear Schrodinger equation of quantum mechanics with the Poisson equation of Newtonian mechanics. In this work, we will investigate the effects of a cosmological constant (dark energy or vacuum fluctuation) on the Schrodinger-Newton system, by modifying the Poisson equation through the addition of a new term. The corresponding Schrodinger-Newton-? system cannot be solved exactly, and therefore for its study one must resort to either numerical or semianalytical methods. In order to obtain a semianalytical solution of the system we apply the Adomian Decomposition Method, a very powerful method used for solving a large class of nonlinear ordinary and partial differential equations. Moreover, the Adomian series are transformed into rational functions by using the Pade approximants. The semianalytical approximation is compared with the exact numerical solution, and the effects of the dark energy on the structure of the Newtonian quantum system are fully investigated.