The stability of the skyrmion magnetization configurations in ultrathin circular magnetic nanodots is calculated analytically and simulated micromagnetically considering realistic magnetic and geometrical dot parameters taken from recent experiments. The chiral Dzyaloshinskii-Moriya exchange interaction (DMI) is taken into account as an interfacial energy term. The diagram of single Néel skyrmion stability/metastability, skyrmion sizes and magnetization profiles are found as a function of the magnetic and geometrical dot parameters. It is shown that most of skyrmions are unstable or metastable and the stability region where the Néel skyrmion is the ground state is small. The Dzyaloshinskii criterion for the instability of ferromagnetic state in bulk ferromagnets with respect to the increasing DMI strength should be essentially modified to describe the stability of magnetic skyrmions in 2D systems like ultrathin ferromagnetic films and nanodots.