Sen. Ted Cruz of Texas says he’s not surprised President Obama took an apparent swipe at him when Mr. Obama on Monday said a “religious test” for admitting Syrian refugees isn’t American.

“It’s not surprising that President Obama’s attacking me personally,” Mr. Cruz said in an interview with CNN. “I’ll tell you what’s shameful, is that we have a president who after seven years still refuses to utter the words ‘radical Islamic terrorism.’ “

Mr. Cruz, a 2016 GOP presidential candidate, has said persecuted Christians should be allowed to seek refuge in the United States, but not Muslims.

Though he didn’t mention Mr. Cruz by name, Mr. Obama went after that stance on Monday.

“When I hear folks say that well, maybe we should just admit the Christians but not the Muslims, when I hear political leaders suggesting that there would be a religious test for which a person who’s fleeing from a war-torn country is admitted, when some of those folks themselves come from families who benefitted from protection when they were fleeing political persecution — that’s shameful,” Mr. Obama said at a press conference in Turkey. “That’s not American. That’s not who we are. We don’t have religious tests to our compassion.”

Mr. Cruz’s father fled Cuba and settled in Canada before eventually ending up in the United States.

Mr. Cruz said if his father were part of a movement like “radical Islamists” that “promotes murdering anyone who doesn’t share your extreme faith or forcibly converting them,” such an attitude would make sense.

“What Barack Obama and Hillary Clinton are proposing is that we bring to this country tens of thousands of Syrian Muslim refugees. I have to say particularly in light of what happened in Paris, that’s nothing short of lunacy,” Mr. Cruz said.

Asked about protecting Muslim victims, Mr. Cruz said: “There is no doubt that millions of people are suffering from the rise of radical Islamic terrorism — Christians are suffering, Jews are suffering, and other Muslims are suffering.”

“What I’m saying is Syrian Muslim refugees should be resettled in the Middle East in majority Muslim countries,” he said.

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