Yes they did, and these are called "naturalization" laws.
if Ted Cruz was issued a Certificate of Naturalization then you are correct. If he wasn’t then you are incorrect.
As I pointed out earlier, naturalization numbers are never issued to children who receive derivative citizenship. Even in cases of actual bonified sworn in, Citizenship Test, Alien naturalization, "naturalized" citizens, they don't give numbers to the Children. The Children acquire "Derivative" citizenship from the naturalized parent(s).
Birth of U.S. Citizens Abroad
A child born abroad to a U.S. citizen parent or parents may acquire U.S. citizenship at birth if certain statutory requirements are met. The child’s parents should contact the nearest U.S. embassy or consulate to apply for a Consular Report of Birth Abroad of a Citizen of the United States of America (CRBA) to document that the child is a U.S. citizen. If the U.S. embassy or consulate determines that the child acquired U.S. citizenship at birth, a consular officer will approve the CRBA application and the Department of State will issue a CRBA, also called a Form FS-240, in the child’s name.
According to U.S. law, a CRBA is proof of U.S. citizenship and may be used to obtain a U.S. passport and register for school, among other purposes.
The child’s parents may choose to apply for a U.S. passport for the child at the same time that they apply for a CRBA. Parents may also choose to apply only for a U.S. passport for the child. Like a CRBA, a full validity, unexpired U.S. passport is proof of U.S. citizenship.
Parents of a child born abroad to a U.S. citizen or citizens should apply for a CRBA and/or a U.S. passport for the child as soon as possible. Failure to promptly document a child who meets the statutory requirements for acquiring U.S. citizenship at birth may cause problems for the parents and the child when attempting to establish the child’s U.S. citizenship and eligibility for the rights and benefits of U.S. citizenship, including entry into the United States. By law, U.S. citizens, including dual nationals, must use a U.S. passport to enter and leave the United States.
Consular Report of Birth Abroad (CRBA, or Form FS-240)
If you are a U.S. citizen and have a child overseas, you should report his or her birth as soon as possible so that a Consular Report of Birth Abroad can be issued as an official record of the child’s claim to U.S. citizenship. Report the birth of your child abroad at the nearest U.S. embassy or consulate. Check the American Citizens Services portion of the webpage for the nearest Embassy or Consulate in the country where your child was born for further instructions about how to apply for a CRBA. Please note:
A Consular Report of Birth Abroad of a U.S. citizen is only issued to a child who acquired U.S. citizenship at birth and who is generally under the age of 18 at the time of the application.
The U.S. embassy or consulate will provide one original copy of an eligible child’s Consular Report of Birth Abroad of a U.S. Citizen.
A more secure Consular Report of Birth Abroad of a U.S. Citizen was introduced in January 2011. This new CRBA has been updated with a variety of state of the art security features, and is printed centrally in the United States. U.S. embassies and consulates no longer print CRBAs locally, but you still must apply there. The central production was initiated to ensure uniform quality and reduce vulnerability to fraud. The previous version of the CRBA continues to be valid proof of U.S. citizenship.
You may replace, amend or request multiple copies of a Consular Report of Birth Abroad of a U.S. Citizen at any time.
Persons who acquired U.S. citizenship or U.S. nationality at birth in one of the following current or former territories or outlying possessions of the United States during relevant time periods are not eligible for a Consular Report of Birth Abroad of a U.S. Citizen because such persons are not considered to have been born abroad. Individuals born in these locations during the relevant times may establish acquisition of U.S. citizenship or non-citizen nationality, based upon the applicable agreement or statute, by producing their birth certificate issued from the local Vital Records Office along with any other evidence required to establish acquisition:
Puerto Rico
U.S. Virgin Islands American Samoa
Guam
Swains Island
The Panama Canal Zone before October 1, 1979
The Commonwealth of the Northern Mariana Islands after January 8, 1978 (8PM EST)
The Philippines before July 4, 1946
Other Citizenship Documents Issued to U.S. Citizens Born Abroad
Certification of Report of Birth (DS-1350)
As of December 31, 2010, the Department of State no longer issues Certifications of Reports of Births (DS-1350). All previously issued DS-1350s are still valid for proof of identity, citizenship and other legal purposes.
Certificate of Citizenship issued by U.S. Citizenship and Immigration Services (USCIS)
A person born abroad who acquired U.S. citizenship at birth but who is over the age of 18 (and so not eligible for a CRBA) may wish to apply for a Certificate of Citizenship to document acquisition pursuant to 8 U.S.C. 1452.
Visit USCIS.gov for further information.