CHICAGO, Aug. 5 -- The company that built its American franchise on the promise of the affordable meal now finds itself unable to hold its own customers. McDonald's named a new president for its United States operations Monday after second-quarter data showed domestic same-store sales rising by the smallest margin in a year, laying bare the precarious position of a brand caught between its working-class identity and the reality of an American consumer who has quietly begun counting what a meal is worth.

McDonald's earned $2.36 billion in net income during the second quarter, or $3.32 per diluted share, up from $2.25 billion, or $3.14 per share, in the same period a year ago. Excluding one-time charges, adjusted earnings reached $3.38 per...