Context. The new census of massive stars in the Carina Nebula reveals the presence of 54 apparently single O-type stars in the Car OB1 association, an extremely active star-forming region that hosts some of the most luminous stars of the Milky Way. A detailed spectroscopic analysis of the current most complete sample of O-type stars in the association can be used to inspect the main physical properties of cluster members and test evolutionary and stellar atmosphere models. Aims. Our aim is to carry out a spectroscopic characterization of the census of the apparently single (actual single or SB1) O-type stars in Car OB1, obtaining a reliable distribution of rotational velocities and stellar parameters from high-resolution spectra from the Gaia-ESO survey (GES) and the LiLiMaRlin library database, which is itself fed by spectroscopic surveys such as the high-resolution spectroscopic monitoring survey of Southern Galactic O- and WN-type stars (OWN), the high-resolution, multi-epoch project of Galactic OB stars (IACOB), the northern massive dim stars survey (NoMaDS), and the Calar Alto fiber-fed echelle - binary evolution Andalusian northern survey (CAF & Eacute;-BEANS). Methods. To derive rotational velocities, we used the semi-automatized tool for the line-broadening characterization of OB stars (iacob-broad) which is based on a combined Fourier transform and the goodness-of-fit methodology. To derive the stellar parameters, we used the iacob-gbat tool, FASTWIND stellar models, and astrometry provided by the Gaia third data release. The BONNSAI tool was used to compute evolutionary masses and ages. Results. We performed quantitative spectroscopic analysis for the most complete sample of apparently single O-type stars in Car OB1 with available spectroscopic data. From the high-resolution GES and OWN spectra, we obtain a reliable distribution of rotational velocities for a sample of 37 O-type stars. It shows a bimodal structure with a low velocity peak at 60 km s(-1) and a short tail of fast rotators reaching 320 km s(-1). We also performed a quantitative spectroscopic analysis and derived effective temperature, surface gravity, and He abundance for a sample of 47 O-type stars, now including further stars from the GOSSS database. Radii, luminosities, and spectroscopic masses were also determined using Gaia astrometry. We created the Hertzsprung-Russell diagram to inspect the evolutionary status of the region and confirm the lack of stars close to the zero-age main sequence (ZAMS) from similar to 35-55 M-circle dot. We confirm a very young population with an age distribution peaking at 1 Myr, with some stars close to or even on the ZAMS, and a secondary peak at 4-5 Myr in the age distribution. We confirm the youth of Trumpler 14, which is also the only cluster not showing the secondary peak. We also find a clear trend of evolutionary masses higher than derived spectroscopic masses for stars with an evolutionary mass below 40 M-circle dot .