This is how the SCOTUS cases approach the issue–but I believe a more correct reading of Vattel and the correct approach under natural law is to consider the child to be NBC of the same country as the father, so no naturalization should be required here. (Ted Cruz, on the other hand, had no US father.)
Mar. 21, 2016) — Last week, The Post & Email was contacted by a reader who informed us that he or she was born abroad to U.S.-citizen parents, both of whom were “natural born Citizens.” At the time of the child’s birth, his or her father was serving in the U.S. Army.When the reader contacted us, he or she said:I was born overseas while my father was in the Army. Both my parents are natural born. I am not. When I joined the military in 1974 I had to apply for a Naturalization Certificate. I would have had to have done so a year later anyway when I would have turned 18 if the law requiring me to do so had not been repealed that year. I had to provide my birth certificate and my form FS-240 Report of Birth. I am a Naturalized citizen and at best that is all Ted Cruz is.