Superconductor-ferromagnet heterostructures hosting vortices and skyrmions are a new area of the interplay between superconductivity and magnetism. We study the interaction of a Néel-type skyrmion and a Pearl vortex in thin heterostructures due to stray fields. Surprisingly, we find that it can be energetically favorable for the Pearl vortex to be situated at some nonzero distance from the center of the Néel-type skyrmion. The presence of a vortex-antivortex pair is found to result in the increase of the skyrmion radius. Our theory predicts that a spontaneous generation of a vortex-antivortex pair is possible under some conditions in the presence of a Néel-type skyrmion.