I believe Cruz was registered at birth as an American citizen and was never registered as a Canadian. He did not have to be naturalized as he was a citizen of only the US.
When a baby is born of parents of two different countries, the parents may choose a single nationality or a double nationality for the baby. The baby is considered a “live birth” until the parents fill out the citizenship papers.
My grandson was born in England to an English mother and American father. The parents had a choice of making him only an American citizen or only a British citizen or a dual citizen. They chose for him to be a dual citizen and my son went to the American embassy that day and registered him as American. He has a full British passport and a full American passport. He does not have to be naturalized in either country as he is a full citizen. When he comes to the states, he uses his American passport and lines up with Americans to enter the country. If his mother comes, she has to get in the foreign country line. Certainly, a person with dual citizenship could not run for president or for prime minister in England. He would have to relinquish the citizenship of the other country in order to have full allegiance to the country where he wanted to hold office. If Britain has the same law we do about holding office, he would have to relinquish his American citizenship, then he could run for prime minister.
Until Cruz himself actually says in public that he was never a Canadian citizen, this will keep coming up. I read early on he was only an American citizen but no one will believe that until it is formally documented by him. If he had dual citizenship, he would have to drop his Canadian citizenship so his allegiance would be only to this country.
If his parents listed him only as Canadian and not American, then he would have to be naturalized later as an American when he came here, just like any person from another country could come here and go through the process to become a naturalized citizen.
I won’t get into that “natural born” citizenship argument. One of these days, Cruz will make a statement that will clear this up. I certainly believe he will tell the truth as truth seems to be a virtue he has. If he is a naturalized citizen, we lose the best candidate for president we could have.
Natural citizens can't chose.