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As communication continues to develop, the high freedom and low cost of the communication network environment also make rumors spread more rapidly. If rumors are not clarified and controlled in time, it is very easy to trigger mass panic and undermine social stability. Therefore, it is important to establish an efficient model for rumor propagation. In this paper, the impact of rumor clarifiers on the spread of rumors is considered and fractional order differentiation is introduced to solve the problem that traditional models do not take into account the “anomalous propagation” characteristics of information. A fractional-order Susceptible-Infected-Removal-Clarify (SIR-C) rumor propagation prediction model featuring the clarification mechanism is proposed. The existence and asymptotic stability conditions of the rumor-free equilibrium point (RFEP) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mn>0</mn></msub></semantics></math></inline-formula>; the boundary equilibrium points (BEPs) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mn>2</mn></msub></semantics></math></inline-formula> are also given. Finally, the stability conditions and practical cases are verified by numerical simulations. The experimental results confirm the analysis of the theoretical study and the model fits well with the real-world case data with just minor deviations. As a result, the model can play a positive and effective role in rumor propagation prediction.