Textual-edge Graphs (TEGs), characterized by rich text annotations on edges, are increasingly significant in network science due to their ability to capture rich contextual information among entities. Existing works have proposed various edge-aware graph neural networks (GNNs) or let language models directly make predictions. However, they often fall short of fully capturing the contextualized semantics on edges and graph topology, respectively. This inadequacy is particularly evident in link prediction tasks that require a comprehensive understanding of graph topology and semantics between nodes. In this paper, we present a novel framework - Link2Doc, designed especially for link prediction on textual-edge graphs. Specifically, we propose to summarize neighborhood information between node pairs as a human-written document to preserve both semantic and topology information. A self-supervised learning model is then utilized to enhance GNN's text-understanding ability from language models. Empirical evaluations, including link prediction, edge classification, parameter analysis, runtime comparison, and ablation studies, on four real-world datasets demonstrate that Link2Doc achieves generally better performance against existing edge-aware GNNs and pre-trained language models in predicting links on TEGs.