Supercritical carbon dioxide (SCO2) jets are a promising method to assist drilling, enhance oil–gas production, and reduce greenhouse gas emissions. To further improve the drilling efficiency of SCO2 jet-assisted drilling, organ-pipe nozzles were applied to generate a self-excited oscillation SCO2 jet (SEOSJ). The impact pressure oscillation and rock erosion capability of SEOSJs under both supercritical and gaseous CO2 (GCO2) ambient conditions were experimentally investigated. It was found that the impact pressure oscillation characteristics of SEOSJs produced by organ-pipe nozzles are dramatically affected by the oscillation chamber length. The optimum range of the dimensionless chamber length to generate the highest impact pressure peak and the strongest pressure oscillation is within 7–9. The dimensionless pressure peak and the pressure ratio decreases gradually with increasing pressure difference, whereas the pressure oscillation intensity increases with increasing pressure difference and the increasing rate decreases gradually. The dominant frequency was observed to decrease monotonically with increasing chamber length but increases with the increase of pressure difference. Moreover, the comparison of impingement characteristics of SEOSJs under different ambient conditions showed that the values of dimensionless peak impact pressure are similar under the two ambient conditions, and the SEOSJ achieves higher pressure oscillation intensity and dominant frequency in SCO2 at the same pressure difference. The rock breaking ability of the SEOSJ is closely related to its axial impact pressure. The erosion depth and mass loss of sandstone caused by the organ-pipe nozzle with the best impact pressure performance is higher than those produced by other nozzles. The SEOSJ results in a deeper and narrower crater in SCO2 than in GCO2 under the same pressure difference. The reported results provide guidance for SEOSJ applications and the design of an organ-pipe nozzle used for jet-assisted drilling.