What does the US Supreme Court say about it?
“There is, however, one clear exception to the statement that there is no national common law. The interpretation of the Constitution of the United States is necessarily influenced by the fact that its provisions are framed in the language of the English common law, and are to be read in the light of its history. The code of constitutional and statutory construction which therefore is gradually formed by the judgments of this Court, in the application of the Constitution and the laws and treaties made in pursuance thereof, has for its basis so much of the common law as may be implied in the subject, and constitutes a common law resting on national authority.”
http://supreme.justia.com/cases/federal/us/124/465/case.html
” The language of the Constitution and of many acts of Congress could not be understood without reference to the common law.”
http://supreme.justia.com/cases/federal/us/91/270/case.html
Yes, you are, indeed, correct that a “natural born Citizen” is defined under common law. An Article II “natural born Citizen” “[a]t common law, with the nomenclature of which the framers of the Constitution were familiar” were “all children, born in a country, of parents who were its citizens.” Minor v. Happersett, 88 U.S. 162, 167-68 (1875); U.S. v. Wong Kim Ark, 169 U.S. 649, 679-80 (1898) (citing and quoting Minor).