In this paper, a constitutive model, based on the creep deformation mechanism in P91 steel, under a wide range of stress levels, was established and embedded into finite element software. The accuracy and reliability of the model was verified by comparing the simulation of uniaxial creep tensile test results and the experimental data under different stress levels for P91 steel at 600 °C. The creep crack growth behavior of P91 steel, under a wide range of stress levels was simulated using a ductility-exhaustion-based damage model, combined with the stress-dependent creep ductility model, and the predicted creep crack growth (CCG) rates were compared with the experimental data. Finally, the established model was used to predict the CCG behavior for the pressurized pipes with axial surface cracks. The results show that the constitutive model, established on the basis of the creep deformation mechanism, agrees better with the experimental data than other constitutive models. The CCG rate varies at different direction angles θ for the axial surface cracks. The direction angle θ corresponding to the maximum creep crack length is about 33°, when the internal pressure exceeds 10 MPa. The initial crack shape (a0/c0) = 1, and it does not change with different initial crack depth ratios (a0/t). The established constitutive model can be well used in CCG life analyses and designs of high-temperature structures.