We study the nonparaxial diffraction of a Gaussian vortex beam with initial radial polarization and an arbitrary integer topological charge n. Analytical relationships for the radial, azimuthal, and longitudinal components of the E-vector are deduced. At $n=0$ , the azimuthal component of the field equals zero, with the radial and axial components becoming coincident with the relationships reported in [ J. Opt. Soc. Am. A 26, 1366 (2009) ]. At any $n>1$ , the vortex beam intensity on the optical axis equals zero, whereas at $n=1$ $(-1)$ an intensity peak is found in the focus. Explicit analytical relationships for a Gaussian vortex beam with initial elliptical polarization are also derived. Relationships that describe the nonparaxial radially polarized Gaussian beam are deduced as a linear combination of the Gaussian vortex beams with $n=1$ $(-1)$ and left- and right-hand circular polarization.