Opinion dynamics on social networks have received considerable attentions in recent years. Nevertheless, just a few works have theoretically analyzed the condition in which a certain opinion can spread in the whole structured population. In this article, we propose an evolutionary game approach for a binary opinion model to explore the conditions for an opinion’s spreading. Inspired by real-life observations, we assume that an agent’s choice to select an opinion is not random but is based on a score rooted from both public knowledge and the interactions with neighbors. By means of coalescing random walks, we obtain a condition in which opinion A can be favored to spread on social networks in the weak selection limit. We find that the successfully spreading condition of opinion A is closely related to the basic scores of binary opinions, the feedback scores on opinion interactions, and the structural parameters including the edge weights, the weighted degrees of vertices, and the average degree of the network. In particular, when individuals adjust their opinions based solely on the public information, the vitality of opinion A depends exclusively on the difference of basic scores of A and B. When there are no negative (positive) feedback interactions between connected individuals, we find that the success of opinion A depends on the ratio of the obtained positive (negative) feedback scores of competing opinions. To complete our study, we perform computer simulations on fully connected, small-world, and scale-free networks, respectively, which support and confirm our theoretical findings.