"I would encourage you, Mr. President, if you want to insult me, come back and insult me to my face," Cruz said of Obama, who is in the Philippines.
"Let's have a debate on Syrian refugees right now," he added. "We can do it anywhere you want. I'd prefer it in the United States and not overseas where you're making the insults. It's easy to toss a cheap insult when no one can respond."
On Monday, Obama didn't mention Cruz by name, but blasted his proposal to bar Muslim Syrian refugees from the U.S. and allow in only Christian Syrians.
"When I hear political leaders suggesting that there would be a religious test for which a person who is fleeing from a war-torn country is admitted, when some of those folks themselves come from families who benefited from protection when they were fleeing political persecution, that's shameful, that's not American," Obama said
as he was attending a meeting of the G20 in Antalya, Turkey.
Obama stepped up his attack on Republicans Wednesday during a stop in the Philippines — Tuesday night, U.S. time — and said they were afraid of "widows and orphans."
"Apparently, they are scared of widows and orphans coming into the United States of America," the President said, apparently referring to Chris Christie saying he would not even accept into the U.S. a 5-year-old Syrian orphan. "At first, they were too scared of the press being too tough on them in the debates. Now they are scared of 3-year-old orphans. That doesn't seem so tough to me."
Cruz spoke after a meeting of Republican lawmakers at the Capitol Hill Club where one of his newest supporters in Iowa, Rep. Steve King, pitched his colleagues on why the Texas senator is the best option for retaking the White House.
When Cruz was asked by a reporter about Obama's comments, he called them an attack on him and others who are worried about letting in Syrian refugees who could carry out terror attacks.
"It is utterly un-befitting of a President to be engaging in those kinds of personal insults, attacks," Cruz said, charging that Obama was "belittling the Republican field."
Meanwhile, Republican contender Rick Santorum also knocked Obama Wednesday for his attacks on the GOP candidates, saying that the President "needs to get a grip on reality right now!"
"Christians are being beheaded and crucified by these savages. People of good conscience who want to protect Christians are not recruiting tools for ISIL," Santorum said in a statement. "While I disagree that accepting Christian refugees is in the long-term security interest of the United States and long-term stability interest of the Middle East, the intent of those who do should never be vilified."