The latest source catalog of the Fermi-LAT telescope contains more than 7000 gamma-ray sources at GeV energies, with the two dominant source classes thought to be blazars and rotation-powered pulsars. Despite continuous follow-up efforts, around 2600 sources have no known multiwavelength association. Our target is the identification of possible (young and recycled) pulsar candidates in the sample of unassociated gamma-ray sources via their characteristic X-ray and gamma-ray emission. To achieve this, we cross-match the Fermi-LAT catalog with the catalog of X-ray sources in the western Galactic hemisphere from the first four all-sky surveys of eROSITA on the SRG (Spektrum-Roentgen-Gamma) mission. We complement this by identifying X-ray counterparts of known pulsars detected at gamma-ray and radio energies in the eROSITA data. We use a Bayesian cross-matching scheme to construct a probabilistic catalog of possible pulsar-type X-ray counterparts to Fermi-LAT sources. Our method combines the overlap of X-ray and gamma-ray source positions with a probabilistic classification (into pulsar and blazar candidates) of each source based on its gamma-ray properties and a prediction on the X-ray flux of pulsar- or blazar-type counterparts. Finally, an optical and infrared counterpart search is performed to exclude coronally emitting stars and active galactic nuclei from our catalog. We provide a catalog of our prior gamma-ray-based classifications of all 2600 unassociated sources in the Fermi-LAT catalog, with around equal numbers of pulsar and blazar candidates. Our final list of candidate X-ray counterparts to suspected new high-energy pulsars, cleaned for spurious detections and sources with obvious non-pulsar counterparts, contains around 900 X-ray sources, the vast majority of which lies in the 95% gamma-ray error ellipse. We predict between 30 and 40 new pulsars among our top 200 candidates, with around equal numbers of young and recycled pulsars. This candidate list may serve as input to future follow-up campaigns, looking directly for pulsations or for the orbital modulation of possible binary companions, where it may allow for a drastic reduction in the number of candidate locations to search. We furthermore detect the X-ray counterparts of 15 known rotation-powered pulsars, which were not seen in X-rays before.