The computational finite element model of contact between a steel sphere and a composite plate was developed in the ANSYS software package. The model includes geometric description of the contact, materials management, computational mesh generation using the SOLID186 and SOLID187 finite elements, as well as the surface-to-surface contact model using the CONTA174 and TARGET170 contact elements. The paper identifies influence of the rigid sphere radius on the woven composite structure and shows areas of the local composite destruction. Satisfactory compliance was obtained between the numerical calculation results and full-scale study of a contact between rigid indenter of different radii and the fiberglass laminate. The safety factor values are provided according to four criteria. The developed model of contact between the indenter and the composite plate made it possible to obtain stress fields in the carbon-plastic composite material with the laying layers [45°, –45°]n and [45°, 90°, –45°, 0°]n. Destruction in layers based on the numerical calculation results was analyzed using various stress criteria. Carbon composite had more destroyed layers due to low strength in the transverse direction. Destruction zone in the carbon composite was larger than in the fiberglass composite, and the pattern [45°, –45°]n had a smaller damage area compared to [45°, 90°, –45°, 0°]n.