Not necessarily. According to Vattel, Law of Nations, Must be born of TWO Citizens to be Natural Born and also born on the Soil of the U.S.
A tree is a plant, but a plant is not necessarily a tree.
The founders of the USA wrote the constitution in 1787 and the first naturalization law in 1790. In both documents they use the phrase “natural born citizen”, and it is very clear that by that 1790 statute Ted Cruz would be considered a “natural born citizen” by the founders of the USA.
Emer de Vattel was a Swiss legal expert, and not a founder of the USA.
Donald Trump has made a lot of conservative Republicans very annoyed with this desperate looking and uncalled for attack on Ted Cruz. The last time they had a tiff Cruz provoked him with a comment at a private meeting, but this time Trump is just trying to bully a fellow Republican, and mislead anyone gullible enough to believe his attack. You can do better Mr Trump, and I hope you will.
And Vattel, as interesting as he is, is not US law, and never has been.
The Act also establishes the United States citizenship of certain children of citizens, born abroad, without the need for naturalization: "the children of citizens of the United States that may be born beyond Sea, or out of the limits of the United States, shall be considered as natural born Citizens".
This act was indeed superceded by subsequent acts in 1795 and 1802 and many others up until today, but I think the framers thought they had defined natural born citizen sufficiently to not have to repeat it in every law.
The U.S. didn't even exist when Vattel wrote his book.