Graph Neural Networks (GNNs) have achieved remarkable results in various tasks. Recent studies reveal that graph backdoor attacks can poison the GNN model to predict test nodes with triggers attached as the target class. However, apart from injecting triggers to training nodes, these graph backdoor attacks generally require altering the labels of trigger-attached training nodes into the target class, which is impractical in real-world scenarios. In this work, we focus on the clean-label graph backdoor attack, a realistic but understudied topic where training labels are not modifiable. According to our preliminary analysis, existing graph backdoor attacks generally fail under the clean-label setting. Our further analysis identifies that the core failure of existing methods lies in their inability to poison the prediction logic of GNN models, leading to the triggers being deemed unimportant for prediction. Therefore, we study a novel problem of effective clean-label graph backdoor attacks by poisoning the inner prediction logic of GNN models. We propose BA-Logic to solve the problem by coordinating a poisoned node selector and a logic-poisoning trigger generator. Extensive experiments on real-world datasets demonstrate that our method effectively enhances the attack success rate and surpasses state-of-the-art graph backdoor attack competitors under clean-label settings. Our code is available at https://anonymous.4open.science/r/BA-Logic