WASHINGTON—President Barack Obama, in a rare display of anger, called "sanctimonious" and "purist" the Democratic critics of his tax-cut deal with Republicans, showing the strains of a president caught between his liberal base and the more centrist independent voters who helped him win the White House.

The outburst came at the end of an afternoon news conference where the president ardently defended not only the tax-cut deal but his broader record. Mr. Obama pleaded for support from his core constituents, even as he chastised...