To better understand the creep mechanical behavior of salt rock under very long-term tests and more accurately predict the long-term deformation of the salt caverns, a series of long-term uniaxial creep tests lasting for 21000 h (875 days) were carried out. The results show that the creep strain and creep rate of salt rock increase with the increase of axial stress. The initial creep stage under lower stresses lasted for 4.3–8.3 months, which was much higher than that under high stresses. Predicting the true creep rate of the salt cavern based on the test results of the creep rate in this paper is more accurate than the existing research results. A new nonlinear creep-damage constitutive model is proposed by introducing the nonlinear damage body and using the fractional derivative theory. Using the presented model and the classic creep constitutive models to fit the experimental data, it is found that the presented model best characterizes the creep evolution characteristics of salt rock. The fitting function obtained from the experimental data at different times predicts the final creep results, revealing that the longer the test duration, the more accurate the predicted results. The research in this paper provides the necessary basis for intensive investigation and prediction of the long-term creep mechanical behavior of salt storage caverns.