We investigated the effectiveness of nudge-based messages in promoting the download of contact tracing apps, particularly in the context of low uptake due to serious distrust in the product. Even in the presence of distrust towards the app, promoting downloads can be a beneficial means of preventing infection spread at a stage before pharmacological interventions, such as vaccines or therapeutic drugs, are established. Two studies were conducted with Japanese residents who had not yet downloaded any contact tracing apps. Study 1, based on smartphone location data, targeted 2690 individuals who had gone out despite public instructions to stay at home. Study 2 targeted 4126 individuals whose web-search behavior could be tracked. Nudge-based messages did not increase app downloads in either study. In Study 1, where participants were considered non-cooperative, these messages also did not enhance acceptability such as willingness to accept and intention to download the apps. Conversely, in Study 2, a more representative sample, nudge-based messages emphasizing altruism, economic losses, and medical losses increased app acceptability, although they did not increase searches related to the app's keywords.