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It took an act of Congress (the particular law in place at the time of his birth) in order for him to be a citizen. The U.S. Constitution doesn't grant citizenship to those born in foreign lands whether to citizen parent(s) or not.

Congress only has the power of naturalization. Although, they did try back in 1790 to legislate those born to citizen parents overseas, to be natural born Citizen. But they repealed that in 1795.

The Congress has, of course, changed the law many times:

U.S. Immigration Laws Over Time

Sen. Ted Cruz, assuming he was born in Canada in 1970 to a U.S. citizen mother, would have his citizenship status governed by the 1965 Immigration and Nationality Act, a.k.a. the Hart-Cellar Act.

Cruz was presumably naturalized whenever it was that his mother (most likely) filed the appropriate documents (similar to those) as described on the Naturalization Services page. Since he wouldn't have a birth certificate from a Hospital (or attending/certifying medical personnel) in the U.S., his mother would have needed to apply if she wanted/needed proof of his U.S. Citizenship - N-600. Application for Certificate of Citizenship or applied for a passport for him.

The difference here, is that those who want/need to prove their U.S. citizenship after being born in a foreign country (like Cruz), needed to apply for either a passport or use the N-600 form to lawfully enter the country permanently. Persons born in the U.S. obviously don't need to do that.

So far as I know, nobody in the public domain has seen his birth documents. We are simply taking his word at this point that he is a "citizen"...obtained through the powers and discretion of Congressional naturalization.

“So far as I know, nobody in the public domain has seen his birth documents. We are simply taking his word at this point that he is a “citizen”...obtained through the powers and discretion of Congressional naturalization.”

Barack Obama has never seen his vault copy birth vital records but those records have been seen by three persons in the public domain: Dr. Chiyome Fukino, former Director of the Hawaii Department of Health, Dr. Alvin T. Onaka, Hawaii Registrar of Vital Statistics, and Loretta Fuddy, current Director of the Hawaii Department of Health. All three have issued official statements of verification for the original birth records and those statements have been entered as exhibits in courts of law.
The courts and Congress have taken their words for it, not Obama’s.
For example: “The state of Hawaii has said that the President was born there, that’s good enough for me.”—Rep. John Boehner, Speaker of the House.

Ten courts have ruled explicitly or implicitly that Barack Obama is a natural born citizen. No court has ruled that he is not a natural born citizen.

For example: Allen v Obama, Arizona Superior Court Judge Richard E. Gordon: “Arizona courts are bound by United States Supreme Court precedent in construing the United States Constitution, and this precedent fully supports that President Obama is a natural born citizen under the Constitution and thus qualified to hold the office of President. Contrary to Plaintiff’s assertion, Minor v. Happersett, 88 U.S. 162 (1874), does not hold otherwise.”—Pima County Superior Court, Tuscon, Arizona, March 7, 2012