The particular case where a US citizen gives birth to their child in another country has not been ruled on by the courts. To my logic and that I discern from the founders, a US citizen giving birth to their child elsewhere is a naturalized US citizen.
The Illinois Board of Elections has ruled that he is a natural born citizen. This author thinks, "Congress and not the Supreme Court should be the final legal judge of Cruz's eligibility."
"Yes, Ted Cruz is a ‘natural born citizen’"
The Supreme Court has long recognized that two particularly useful sources in understanding constitutional terms are British common law and enactments of the First Congress. Both confirm that the original meaning of the phrase “natural born Citizen” includes persons born abroad who are citizens from birth based on the citizenship of a parent.
As to the British practice, laws in force in the 1700s recognized that children born outside of the British Empire to subjects of the Crown were subjects themselves and explicitly used “natural born” to encompass such children.
No doubt informed by this longstanding tradition, just three years after the drafting of the Constitution, the First Congress established that children born abroad to U.S. citizens were U.S. citizens at birth, and explicitly recognized that such children were “natural born Citizens.” The Naturalization Act of 1790 provided that “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: Provided, That the right of citizenship shall not descend to persons whose fathers have never been resident in the United States . . . .” ('On the Meaning of “Natural Born Citizen”'. Mar 11, 2015. Neal Katyal & Paul Clement. Harvard Law Review)
"Re-examining the Constitution's Presidential Eligibility Clause"