To research the creep behavior of salt rock, the uniaxial and triaxial creep experiments were carried out under different deviatoric stresses and same confining pressure or different confining pressures and same deviatoric stress. The experiment results showed that the analysis on the creep damage of salt rock should consider the influence of confining pressure and deviatoric stress. By introducing the damage variable to the Norton Power law of salt rock, a new constitutive law of salt rock was presented. The damage equivalent stress was the function of the confining pressure and the deviatoric stress in the new constitutive law. The constitutive law can embody the steady creep and accelerative creep of salt rock, in which the parameters are obtained by simulating the experimental data of salt rock in some salt mine. The theoretic results are in good agreement with the experimental results. The new constitutive law was embedded in the three dimension numerical codes. The natural gas storage in the salt mine was analyzed by the new constitutive law. The evolvement law of damage bound and the displacement law around the natural gas storage were studied under the minimal natural gas storage pressure. The continuous operating time of the natural gas storage was also analyzed under the pressure. The research conclusions are expected to be applied in the natural gas storage projects in salt rock layer.