Sen. Ted Cruz, R-Texas (AP Photo/Paul Vernon)

Responding to Ben Carson's opposition to a Muslim president, Sen. Ted Cruz (R-Tex.) said it would be unconstitutional to prohibit someone from being president because of their religion. But he also said the Obama administration’s plan to allow tens of thousands of Muslim refugees from Syria into the U.S. is “crazy.”

Carson, one of Cruz's opponents in the crowded GOP presidential primary field, has said he "would not advocate putting a Muslim in charge of this nation;" Cruz said that the Constitution does not allow a specific religion to be a qualification for the presidency.

"The Constitution specifies there shall be no religious test for public office, and I'm a constitutionalist," Cruz, who is seeking the GOP presidential nomination, said on "Iowa Press" on Iowa Public Television, in an interview taped Sunday.

Carson, speaking on "Meet The Press" Sunday, said that he does not believe Islam is consistent with the Constitution.

[Read: Ben Carson says he would not support a Muslim for president]

Sen. Lindsey Graham (R-S.C.), who is also running for the Republican nomination, said on Fox News that "America is an idea not owned by a particular religion, race, or anything else."

Cruz made reference to a Donald Trump rally in New Hampshire last week, where Trump did not correct an attendee who said President Obama is Muslim and not American.

[Read: Trump doesn’t correct rally attendee who says Obama is Muslim and ‘not even an American’]

The press, Cruz said, has "loved" asking questions about the incident.

"My view .. the president's faith is between him and God," Cruz said. "What I'm going to focus on is his public policy record and his public policy record has been atrocious."

Cruz said that "when it comes to Christians ... religious liberty is under assault in this country" under the Obama administration. Cruz also said he believes it is dangerous to let a tide of refugees from Syria into the United States. The Obama administration announced Sunday that 15,000 more refugees will be settled in the U.S. next  year.

[Read: 15,000 more refugees to be resettled in U.S. next year]

"I think the president's proposal is nothing short of crazy," Cruz said. "We do not need to be bringing in tens of thousands of Muslim refugees from Syria."

The Texas Republican said that many of the refugees are young men and cited concerns that some may be aligned with the Islamic State. They should be resettled in other Middle Eastern countries and that the United States could provide financial assistance to other countries taking them in, he said.

Cruz said that allowing Christian refugees into the U.S. is a different story because they are not a terrorist threat.

"I think the Christians are a very different circumstance because Christians are  being persecuted," Cruz said, "And the Obama administration has abandoned Middle East Christians."