Red supergiants (RSGs) are essential to understanding the evolution and the contribution to the interstellar medium of massive stars. However, the number of identified RSGs within the Milky Way is still limited mainly due to the difficulty of measuring stellar extinction and distance. The release of approximately one million RVS spectra in Gaia DR3 presents new opportunity for identifying Galactic RSGs, because the equivalent width of the calcium triplet lines (EW(CaT)) in the spectra is an excellent indicator of stellar surface gravity. This work uses the RVS spectra with signal-to-noise ratio (SNR) greater than 100 to search for the Galactic RSGs. The dwarf stars and red giants are removed and the RSG candidates are selected by the location in the EW(CaT) vs. BP-RP diagram. The early-type RSG candidates (K0-M2) are then identified by BP-RP>1.584 and EW(CaT)>1.1nm. To identify late-type RSG candidates (after M2), the criteria of the average equivalent widths of TiO in the XP spectra (EW(TiO)) > 10 nm, the color index K-W3<0.5 and the period-amplitude sequence from Gaia DR3 LPV catalog are further applied to reduce the contamination of late-type red giants and asymptotic giant branch stars. This method yields 30 early-type (K0-M2) and 6196 late-type (after M2) RSG candidates, which is a significant increase to the present Galactic RSG sample. The application of this approach to the spectra with SNR>50 results in 48 early-type and 11491 late-type RSG candidates. This preliminary analysis paves the way for more extensive research with Gaia DR4 when larger spectral datasets are expected to significantly enhance our understanding of Galactic RSG populations.