Evaluating the creep deformation process of heat-resistant steels is important for improving the energy efficiency of power plants by increasing the operating temperature. There is an analysis framework that estimates the rupture time of this process by regressing the strain-time relationship of the creep process using a regression model called the creep constitutive equation. Because many creep constitutive equations have been proposed, it is important to construct a framework to determine which one is best for the creep processes of different steel types at various temperatures and stresses. A Bayesian model selection framework is one of the best frameworks for evaluating the constitutive equations. In previous studies, approximate-expression methods such as the Laplace approximation were used to develop the Bayesian model selection frameworks for creep. Such frameworks are not applicable to creep constitutive equations or data that violate the assumption of the approximation. In this study, we propose a universal Bayesian model selection framework for creep that is applicable to the evaluation of various types of creep constitutive equations. Using the replica exchange Monte Carlo method, we develop a Bayesian model selection framework for creep without an approximate-expression method. To assess the effectiveness of the proposed framework, we applied it to the evaluation of a creep constitutive equation called the Kimura model, which is difficult to evaluate by existing frameworks. Through a model evaluation using the creep measurement data of Grade 91 steel, we confirmed that our proposed framework gives a more reasonable evaluation of the Kimura model than existing frameworks. Investigating the posterior distribution obtained by the proposed framework, we also found a model candidate that could improve the Kimura model.