Cruz has said numerous times in the past few days that Muslim refugees from Syria should be resettled in majority Muslim countries in the Middle East while Christian migrants should be allowed to enter the United States.
"There is no meaningful risk of Christians committing acts of terror. If there were a group of radical Christians pledging to murder anyone who had a different religious view than they, we would have a different national security situation," Cruz said Sunday in Myrtle Beach, S.C.
Cruz has said it is "nothing short of lunacy" to allow Syrian Muslim refugees into the country, particularly after Friday's Paris attacks.
President Obama said in response to Cruz and others Monday it is "shameful" to bar refugees based on religion.
"That's not American. That’s not who we are. We don’t have religious tests to our compassion," Obama said.
Cruz responded on CNN Monday.
"It's not surprising that Obama is attacking me personally," the Texas senator and 2016 Republican presidential candidate told CNN's Dana Bash. "I'll tell you what's shameful is that the president after seven years still refuses to utter the words radical Islamic terrorism [and] claims that somehow [it's] a religious test. [It's] not that at all. It is understanding the nature of the evil we face."
The Texas Republican often mentions how his father fled Cuba for the United States as a young man. Cruz was asked on CNN Monday how he could bar refugees from the United States given his father's story.
"If my father were part of a theocratic and political movement like radical Islam that promotes murdering anyone who doesn't share your extreme faith or forcibly converting them, it would make perfect sense not to let someone in who a embraces political philosophy and theology that says murder the infidels," Cruz said.