Aims. We present and apply a method to identify the stellar content of the ROSAT all-sky survey (RASS). Methods. We performed a crossmatch between the RASS sources and stellar candidates selected from Gaia Early Data Release 3 (EDR3) and estimated stellar probabilities for every RASS source from the geometric properties of the match and additional properties, namely the X-ray to G-band flux ratio and the counterpart distances. Results. A comparison with preliminary detections from the first eROSITA all-sky survey (eRASS1) show that the positional offsets of the RASS sources are larger than expected from the uncertainties given in the RASS catalog. From the RASS sources with reliable positional uncertainties, we identify 28 630 (24.9%) sources as stellar; this is the largest sample of stellar X-ray sources to date. Directly from the stellar probabilities, we estimate the completeness and reliability of the sample to be about 93% and confirm this value by comparing it to the identification of randomly shifted RASS sources, preliminary stellar eRASS1 identifications, and results from a previous identification of RASS sources. Our stellar RASS sources contain sources of all spectral types and luminosity classes. According to their position in the color-magnitude diagram, many stellar RASS sources are young stars with ages of a few 10⁷ yr or binaries. When plotting the X-ray to bolometric flux ratio as a function of the color, the onset of convection and the saturation limit are clearly visible. We note that later-type stars reach continuously higher F_X/F_bol values, which is probably due to more frequent flaring. The color distribution of the stellar RASS sources clearly differs from the unrelated background sources. We present the three-dimensional distribution of the stellar RASS sources that shows a clear increase in the source density near known stellar clusters.