High-quality whole-slide scanning is expensive, complex, and time-consuming, thus limiting the acquisition and utilization of high-resolution histopathology images in daily clinical work. Deep learning-based single-image super-resolution (SISR) techniques provide an effective way to solve this problem. However, the existing SISR models applied in histopathology images can only work in fixed integer scaling factors, decreasing their applicability. Though methods based on implicit neural representation (INR) have shown promising results in arbitrary-scale super-resolution (SR) of natural images, applying them directly to histopathology images is inadequate because they have unique fine-grained image textures different from natural images. Thus, we propose an Implicit Self-Texture Enhancement-based dual-branch framework (ISTE) for arbitrary-scale SR of histopathology images to address this challenge. The proposed ISTE contains a feature aggregation branch and a texture learning branch. We employ the feature aggregation branch to enhance the learning of the local details for SR images while utilizing the texture learning branch to enhance the learning of high-frequency texture details. Then, we design a two-stage texture enhancement strategy to fuse the features from the two branches to obtain the SR images. Experiments on publicly available datasets, including TMA, HistoSR, and the TCGA lung cancer datasets, demonstrate that ISTE outperforms existing fixed-scale and arbitrary-scale SR algorithms across various scaling factors. Additionally, extensive experiments have shown that the histopathology images reconstructed by the proposed ISTE are applicable to downstream pathology image analysis tasks.