Cruz, speaking to reporters in Baton Rouge, La., noted that Branstad's son has thrown his support elsewhere, hosting fundraisers for New Jersey Gov. Chris Christie.
Brandstad's son "is leading the effort of lobbyists and Democrats to run attack ads against our campaign because they're supporting a more moderate candidate," he said. "They’re entitled to do that and they’r entitled to engage in whatever attacks they want. It's the nature of politics."
Donald Trump, with whom Cruz had an odd political bromance, has turned on Cruz and now repeatedly questions his ability to run for president. At a rally in New Hampshire on Monday, Trump brought up the issue again.
"Whether you like it or not, Ted has to figure it out because we can't be having a nominee — if he got the nod, I think we're going to win very solidly — but if you get the nomination you can't have the person that gets the nomination be sued like Hillary might be sued." (Trump was referring to Hillary Clinton's troubles over using a private email server when she was secretary of state.)
Trump has said that Cruz should go to court and seek a judgment on whether he can be president, an assertion he stated again Monday.
"I think he can go in for some kind of a judiciary proceeding and — look, it doesn't matter what he does. You can't have a nominee who is going to be subject to be thrown out as nominee. You just can't do it. So you're going to have to make that decision, folks. It's one of those little decisions. I'm sure Ted is thrilled that I'm helping him out. But I am! I mean, I am!" Trump said.
Cruz was Trump's top target during a rally in Reno, Nev., on Sunday. He asked the audience whether Cruz is eligible to run for president.
"Is he a natural-born citizen?" he asked several thousand people gathered in a ballroom. Members of the crowd shouted back, "No!"
"I don't know," Trump said. "Honestly, we don't know. Who the hell knows."
Republican National Committee Chairman Reince Priebus punted when asked whether Cruz is eligible to run for president.
"I’m not going to get in the middle of all these candidate issues. It’s a bad place for me to be," he told Time Magazine. "I’ll let all these folks argue about this stuff, and I’m going to stay out of it."
In Baton Rouge, Cruz said the attacks are a sign of his strength.
"Three to four weeks ago just about every Republican in the field was attacking Donald Trump. Today just about every Republican in the field is attacking me," Cruz said. "That kind of suggests maybe something has changed in the race."
Cruz was born in Calgary, Alberta, in 1970 to an American mother and Cuban-born father. He has said he is an American citizen by virtue of his mother's U.S. citizenship and has renounced his Canadian citizenship. Legal scholars have said Cruz meets the requirement of natural-born citizenship, though it is untested in the courts.
The issue of Cruz's birthplace didn't take center stage in the primary until Trump, who had a detente with Cruz for months, told The Washington Post it could be a "precarious" issue for the Republican Party if Cruz is its nominee. It has led others to question Cruz's eligibility, including Sen. Rand Paul (R-Ky.), who is also running for president, and Cruz's longtime Senate nemesis John McCain (R-Ariz.).
McCain told KFYI it is "worth looking into" whether Cruz is eligible to run for president.
When asked how Cruz could run for president if he was born in Canada, McCain answered, "I do not know the answer to that."
McCain was born in the Panama Canal Zone in 1936, and the Senate unanimously passed a resolution declaring him a natural-born citizen when he ran for president in 2008. McCain said the issue is different because the Canal Zone was a territory and U.S. military base, and there was precedent set when Barry Goldwater, who was born in Arizona when it was a territory, ran for president.
"That's different than being born on foreign soil," he said.
"I think there is a question," he said of Cruz. "I am not a constitutional scholar on that but I think it's worth looking into. I don't think it's illegitimate to look into it."
Cruz brushed off Trump's initial comments by tweeting a video clip of a scene from "Happy Days" where a character water skis over a shark; "jumping the shark" has come to represent a gimmick that indicates a decline in quality. He has also said that the media likes to gin up controversies between candidates.
In Iowa last week, Cruz said that he has always been an American citizen.
"I have never breathed a breath of air on this planet when I was not a U.S. citizen. I’ve never been naturalized. It was the process of being born that made me a U.S. citizen. And so the legal issue is straightforward,” he said.
Jenna Johnson in Reno and Jose A. Del Real in Washington contributed to this report.