Pseudo-linear impulsive differential equations in a Banach space are considered. It is assumed that the conditions of a small change in the operator coefficients of the equation are satisfied. Using the method of "frozen" coefficients and the methods of commutator calculus, the problem of global asymptotic stability of a pseudo-linear impulsive differential equation is reduced to the problem of estimating the evolution operator for linear impulsive differential equation with constant operator coefficients. The obtained results are applied for stability study of a nonlinear system of ordinary impulsive differential equations. Lyapunov's direct method is used for estimating the fundamental matrix of the corresponding system of impulsive differential equations with constant coefficients. The stability conditions are formulated in terms of the solvability of certain linear matrix inequalities.