The Galactic flux of cosmic-ray (CR) positrons in the GeV to TeV energy range is very likely due to different Galactic components. One of these is the inelastic scattering of CR nuclei with the atoms of the interstellar medium. The precise amount of this component determines the eventual contribution from other sources. We present here a new estimation of the secondary CR positron flux by incorporating the latest results for the production cross sections of $e^\pm$ from hadronic scatterings calibrated on collider data. All the reactions for CR nuclei up to silicon scattering on both hydrogen and helium are included. The propagation models are derived consistently by fits on primary and secondary CR nuclei data. Models with a small halo size ($\leq 2$ kpc) are disfavored by the nuclei data although the current uncertainties on the beryllium nuclear cross sections may impact this result. The resulting positron flux shows a strong dependence on the Galactic halo. Within the most reliable propagation models, the positron flux matches the data for energies below 1 GeV. We verify that secondary positrons contribute less than $70\%$ of the data already above a few GeV, demonstrating that an excess of positrons is already present at very low energies. At larger energies, our predictions are below the data with the discrepancy becoming more pronounced. Our predictions are provided together with uncertainties due to propagation and hadronic cross sections. In addition to the predictions of positrons, we provide new predictions also for the secondary CR electron flux.