Not unless he filed the proper papers with either the Consulate or the proper government agency in the US.
From the Dallas Morning News in a story about Cruz having dual citizenship:
“Senator Cruz became a U.S. citizen at birth, and he never had to go through a naturalization process after birth to become a U.S. citizen,†said spokeswoman Catherine Frazier. “To our knowledge, he never had Canadian citizenship.â€
The U.S. Constitution allows only a “natural born†American citizen to serve as president. Most legal scholars who have studied the question agree that includes an American born overseas to an American parent, such as Cruz.
(snip)
“Her status made the baby a U.S. citizen at birth. For that, U.S. law required at least one parent who was a U.S. citizen who had lived for at least a decade in the United States.
“She registered his birth with the U.S. consulate, Frazier said, and the future senator received a U.S. passport in 1986 ahead of a high school trip to England.”