Shahid Shafi was drawn to the Republican tenets of small government and secure borders after growing up under a military dictatorship in Pakistan. Now he is under fire from conservative activists who say he isn’t fit for a leadership position in the party because he is Muslim.

Five months after he was named vice chairman of the Republican Party in Texas’ Tarrant County, which includes Fort Worth, a faction including local precinct chairs are seeking to recall him at a vote in January. One described the conflict as a “civil...