In early scenes of “American Sniper,” which is based on the true story of Chris Kyle, we get a taste of what kind of man he is going to be. As a kid growing up in Texas, he goes deer hunting with his father and hits a long-distance target using a rifle almost as big as he is. At school, he pummels a bully who picks on his little brother. At the dinner table, Dad tells the boys there are three kinds of people in the world: wolves, sheep and sheepdogs.

Mr. Kyle becomes a Navy SEAL, assigned to protect U.S. forces in Iraq by...