MOUNTAIN VIEW, Calif.—Kevin Cernekee was still a “Noogler”—Google’s term for a new employee—when his conservative take on political and social issues raised hackles within the search giant.

After several posts on the company’s freewheeling internal message boards in early 2015 rankled some colleagues, he was given an official warning from human resources about conduct deemed disrespectful and insubordinate. Around that time, a senior manager wrote on the boards that he added Mr. Cernekee to a “written blacklist” of employees...