This paper deals with solving the problem of constructing two-level computational models relating the stress-strain state at the microscale to the stress-strain state of the structural component at the macroscale for layered polymer composite materials with unidirectional fiber stacking in a monolayer. The paper presents a two-level computational mechanical model of a ten-layer cross-ply fiberglass-reinforced polymer composite with (0/90) laying patterns. The model is based on microindentation data. The model is constructed using the finite element method. The constructed model has allowed us to predict the elastic properties of the composite and to calculate stresses at the microscale and macroscale levels of a specimen prior to fracture under tensile conditions. The elastic properties of the polymer composite material predicted by the computational model are compared with the properties obtained from real experiments. Calculated for fiberglass, the elasticity modulus and Poisson’s ratio are 31 GPa and 0.11, respectively. Similar elastic properties obtained from real experiments are 28 GPa and 0.10. The technique used in the paper can be applied to computational experiments with an actual structural component made of a multilayer polymer composite in order to determine the stress-strain state at the micro- and macroscale under external mechanical impact on the structure.