MOSCOW—In his spare time, Alexei Pavlovsky became a local coordinator of Total Dictation—a nationwide grammar and spelling competition that gets strong backing from the government. But last month, he got some unexpected state attention: a summons from local prosecutors he says were on the hunt for “grammar Nazis.”

The Kremlin and Russian state media highlighted the dangers of fascism in a push to bolster national pride ahead of the 70th anniversary of the Soviet Union’s victory over Nazi Germany in May. The term was used...