"But it is precisely the Obama administration’s unwillingness to recognize that or ask those questions that makes them so unable to fight this enemy. Because they pretend as if there is no religious aspect to this," he said.
Former Florida governor Jeb Bush also said Sunday that any assistance going to refugees from the Middle East should be concentrated on Christians.
"We should focus our efforts as it relates to refugees on the Christians that are being slaughtered," he told CNN's "State of the Union."
Cruz did not say how he would determine that refugees were Christian or Muslim. He reiterated his assertion that it is "lunacy" to allow Muslim refugees into the United States, asserting that there is no way to know if they are aligned with the Islamic State.
"We can’t roll the dice with the safety of Americans and bring in people for whom there is an unacceptable risk that they could be jihadists coming here to kill Americans," Cruz said. "We just saw in Paris what happens when a country allows ISIS terrorists to come in as refugees and the result can be a horrific loss of life," he said, using an acronym for the Islamic State.
The assertion is a shift for Cruz, who in early 2014 told Fox News that America should allow Syrian refugees to come to the United States. Now he is saying that Muslim refugees should be resettled in majority Muslim countries in the Middle East.
"We have welcomed refugees, the tired huddled masses for centuries. That’s been the history of the United States. We should continue to do so,” he said. “We have to continue to be vigilant to make sure those coming are not affiliated with the terrorists, but we can do that."
Cruz said Sunday that the landscape has changed since early 2014.
"We’ve seen a number of things change. We’ve seen the rise of ISIS, we’ve seen the manifest evil of their terrorism, we’ve seen also the enormous failures of the Obama administration’s intelligence operations," he said, including that U.S. Central Command is "cooking intelligence" to please the White House. Cruz said the administration "does not have the information" to determine who among refugees might be a terrorist.
"Sadly, it appears that at least one of the terrorists who committed these attacks in Paris came as a refugee in Syria," he said. "So what does President Obama and Hillary Clinton want to do? Bring tens of thousands of them to America and put them here."
The issue of Syrian refugees has loomed large during the Republican primary, with concern about the refugees growing among conservatives in recent months. Gov. Bobby Jindal said borders should be sealed. Ben Carson said accepting the refugees is a "suspension of intellect." Donald Trump has promised to kick Syrian refugees out and not let any more in. Sunday, Sen. Marco Rubio (R-Fla.) said the country cannot absorb any more Syrian refugees.
"It's not that we don't want to; it's that we can't because there's no way to background check someone that's coming from Syria," he said on ABC's "This Week with George Stephanopoulos."
Cruz was asked by a woman here if Congress could halt immigration for two years. He did not answer the question, instead stating that Christians should be let in but Muslims should not.
"We need to be working to provide a safe haven for those Christians who are being persecuted and facing genocide, and at the same time we shouldn't be letting terrorists into America," he said.
Lavinia Limon, the president and chief executive of the U.S. Commission for Refugees and Immigrants, said she is surprised the once-nonpartisan cause of helping refugees fleeing violence has become so politicized. She said it takes about three years for refugees to go through stringent security screenings.
“The definition of a refugee is someone fleeing oppression. They’re fleeing terrorism,” she said. “They’ve experienced what happened in Paris on a daily basis.”
Limon said she is dismayed that presidential candidates want to separate refugees by religion.
"That's the same distinction that ISIS makes between Muslims and Christians. It's playing into their hands," she said.
Ken Knight, 54, a heating and cooling technician from Florence, S.C., said he agrees with Bush and Cruz that only Christian refugees should be let into the country.
"I wouldn't bring the Muslims. They cut your head off. You can't trust them. I'm sure there are good ones, but they're like the mob. Once you get in, you can't trust them," he said after a church service Cruz attended in Florence.
Paulette Heckman of Florence said she believes the situation in Syria is creating a humanitarian crisis and feels sympathy for those fleeing. But she is leery of allowing many people into the United States until the nation's security screening systems are made more robust.
"As Christians we do have to be compassionate and empathetic," she said in Florence. "But I just believe as a country we've got to have a system in place to ensure safety to the best of our ability."
Abby Livingston contributed reporting.