"This issue is a non-issue," Cruz told CBS News. "The law is quite clear. The child of a U.S. citizen born abroad is a natural born citizen."
Cruz has perceived this as an attack, although Trump says he's trying to help him.
"The funny thing about politics, it's fairly unusual for your opponents who are running for the same position to be actually trying to help you," Cruz countered.
Cruz has climbed to the top of the Iowa polls by reaching out to evangelicals and social conservatives. In 2012, 57 percent of Republican caucus-goers described themselves as evangelicals, and 47 percent said they were "very conservative."
But Cruz says his strategy does not rely just on Iowa -- or New Hampshire, whose more moderate electorate has not been as welcoming.
"There are a lot of candidates in this race who have to win Iowa. There are a lot of other candidates who have to win New Hampshire," he said. "From our perspective, we don't view any one state as a must win. We going to compete hard and try to win everywhere."
Cruz has recently taken a harder line on immigration than Trump, opposing Trump's willingness to allow deported immigrants to return to the U.S.
"I do not believe that anyone here has come here illegally should be eligible for citizenship," Cruz said.
Cruz faced the reality of that policy Wednesday in Storm Lake, when he met a woman currently protected from deportation by Obama administration executive actions.
He told the woman under a Cruz presidency, she would have to leave, and that "there are human tragedies when people break the law."