On issues such as Social Security, taxes and environmental regulation, the Newt Gingrich of the 1980s and 1990s is proving to be a problem for the 2012 presidential hopeful, who promises to fundamentally transform Washington. From appearing in a television ad advocating global-warming awareness with then-House Speaker Nancy Pelosi to backing the 1986 amnesty for illegal immigrants to his early support for an individual mandate requiring Americans to purchase health insurance, Mr. Gingrich’s past is sprinkled with positions that cause some conservatives heartburn. Mr. Gingrich earlier this year waved off the Pelosi ad as “probably the single dumbest thing I’ve...