The law is straightforward. Under the Constitution, in order to be president, you must be a natural born citizen. U.S. law has been clear from the very first days of this country that the child of a U.S. citizen born abroad is a natural born citizen.
“Despite the happenstance of a birth across the border, there is no question that Senator Cruz has been a citizen from birth and is thus a ‘natural born Citizen’ within the meaning of the Constitution.”
–Harvard Law Review
This is an issue that has come up many times in American history. Senator John McCain was born in Panama, but he is a U.S. citizen because both of his parents were U.S. citizens. He was a citizen by birth. Likewise, George Romney, Mitt Romney’s father, who ran for president in 1968, was born in Mexico. He was also a U.S. citizen by birth because his parents were citizens. And the third example, interestingly enough, is Barry Goldwater. Goldwater was born in Arizona before Arizona was a state, it was just a territory. And so he was a U.S. citizen by birth by virtue of the fact that his parents were citizens.
As to Senator Cruz, his mother, Eleanor Darragh was born in Wilmington, Delaware. She is a citizen by birth, so Ted became an American citizen by virtue of being born to her.
A bipartisan group of former U.S. solicitors general, Neal Katayal and Paul Clement explained the legal case in the Harvard Law Review last year:
There are plenty of serious issues to debate in the upcoming presidential election cycle. The less time spent dealing with specious objections to candidate eligibility, the better. Fortunately, the Constitution is refreshingly clear on these eligibility issues. To serve, an individual must be at least thirty-five years old and a “natural born Citizen.” Thirty-four and a half is not enough and, for better or worse, a naturalized citizen cannot serve. But as Congress has recognized since the Founding, a person born abroad to a U.S. citizen parent is generally a U.S. citizen from birth with no need for naturalization. And the phrase “natural born Citizen” in the Constitution encompasses all such citizens from birth. Thus, an individual born to a U.S. citizen parent — whether in California or Canada or the Canal Zone — is a U.S. citizen from birth and is fully eligible to serve as President if the people so choose.”