Pub.L. 82-414 § 301(a)(7),(b); 66 Stat. 236
http://uscode.house.gov/statviewer.htm?volume=66&page=236#
Based on that, I would say “yes” he does.
A child born abroad to one U.S. citizen parent and one non-citizen parent acquires U.S. citizenship at birth under Section 301(g) of the Immigration and Nationality Act, provided the American parent was physically present in the United States or one of its outlying possessions for the time period required by the law applicable at the time of the child’s birth.
For birth between Dec. 24, 1952, and Nov. 13, 1986, a period of 10 years — five after the age of 14 — is required for physical presence in the United States or one of its outlying possessions to transmit U.S. citizenship to the child.
Since Cruz entered the United States in possession of a United States passport, I presume his mother met the physical presence requirements of the Immigration and Nationality Act.
This is not my opinion, it is the opinion of an immigration judge. see here:
http://www.latimes.com/opinion/readersreact/la-le-0228-saturday-ted-cruz-20150228-story.html