They don't, American constitutional scholars say, because Cruz's mother was a U.S. citizen when he was born. His Cuban father became a citizen in 2005.
The fact that Cruz's mother is Delaware-born satisfies the U.S. constitution's requirement that a president must be a "natural-born U.S. citizen," experts agree.
Last spring, Cruz — dubbed "Canadian Ted" by some of his political foes in Texas — formally gave up his Canadian citizenship.
Nonetheless, the questions about Cruz's legitimacy are rich with irony, since the senator's own father — Rafael, who appears at public events regularly on behalf of his son — once raised doubts about Barack Obama's birthplace.
In 2012, the pastor argued Obama should be sent "back to Kenya."
Obama was born in Hawaii to an American mother and Kenyan father, but those in Cruz's Tea Party base have long believed the U.S. president was actually born in Kenya and therefore ineligible to occupy the White House.
Cruz's years in politics have also shown him to be, essentially, an anti-immigration immigrant. Recently, he's been a fierce opponent to Obama's plans to give amnesty to millions of illegal immigrants.
He once railed against so-called anchor babies — children born to illegal immigrants in the U.S., making them U.S. citizens, ostensibly to help avoid deportation — while claiming he didn't know that being born in Calgary 44 years ago made him Canadian.
Cruz explained it this way: "When I was a kid, my mom told me that if I ever wanted to, I could affirmatively choose to claim Canadian citizenship, but I got a U.S. passport when I was in high school, I never did anything to affirmatively claim citizenship, so I thought that was the end of the matter."
In an interview on CNN two years ago, Cruz said he didn't know he was a Canadian until the Dallas Morning News reported in the summer of 2013 that he had dual citizenship.
For that, media wags in his home state dubbed him a Canadian anchor baby.
CNN's Candy Crowley, meantime, argued Cruz's Canadian citizenship was pretty much the most interesting thing about him.
Following Cruz's official announcement of his run for president on Monday, some news organizations wondered what all the fuss was about regarding his birthplace.
The New Republic argued that if Cruz's political foes try to attack him for his Calgary roots, Cruz has a ready retort.
"Calgary is a fair approximation of a Red State American city, a frigid Dallas," Jeet Heer wrote. "If you can't be born in conservative America, Calgary is about as close to second best as the world has to offer."
Indeed, at a Cruz town hall in Texas in 2013, a Republican voter who still believed Obama was born in Kenya explained why the senator's Calgary birthplace didn't trouble her.
"As far as I'm concerned, Canada is not really foreign soil," Christina Katok opined.
Social media was abuzz on Monday with Canadian Cruz jokes, some of them taking aim at Donald Trump, the most famous so-called Obama birther.
"I'm sure any day now @realDonaldTrump will be asking to see @tedcruz's birth certificate. JK LOL he's white!" tweeted Danny Zuker, writer and executive producer at ABC's "Modern Family."
Trump, meantime, suggested in an interview with Fox News that Cruz's Canadian ties could, indeed, pose problems for his presidential bid — despite what the constitutional experts say.
"It’s a hurdle, somebody could certainly look at it very seriously," he said.
"He was born in Canada ... when we all studied our history lessons, you are supposed to be born in this country, so I just don't know how the courts will rule on this."
Follow Lee-Anne Goodman on Twitter @leeanne25