After more than a year of research and discussion, Facebook Inc. late this summer shelved a project called “Common Ground” that tried to encourage users with different political beliefs to interact in less-hostile ways. One reason: fears the proposed fix could trigger claims of bias against conservatives, according to people familiar with the decision.

The objections were raised by Joel Kaplan, a former White House aide to George W. Bush who has emerged as Facebook’s protector against allegations of political bias—and thus one of its most powerful and controversial executives.

Mr. Kaplan is Facebook’s longtime global policy chief but his remit has expanded considerably in the last two years. He has often been the decisive word internally on hot-button political issues and has wielded his influence to postpone or kill projects that risk upsetting conservatives, said the people familiar with the decisions.

Mr. Kaplan now has a say in whether certain news-feed products get launched, an area that has long been the purview of Facebook’s engineers and product teams.

Mr. Kaplan personifies an unresolved tension within Facebook amid the drumbeat of criticism since the 2016 presidential election: The company has set a goal of minimizing polarization and misinformation, but it is increasingly wary of being perceived as being partisan. Many current and former Facebook insiders argue that the company’s desire to avoid criticism from conservatives prevents it from fully tackling broader issues on the platform.