The presidential hopeful announced his position at a news conference scheduled late Sunday night. He told reporters that he has sent a letter to House Speaker Paul Ryan asking to move a bill that "terminates all ongoing funding for any federal programs that seek to resettle refugees and/or migrants from Syria into the United States, effective immediately."
"Congress I think should defund all the programs that allow all these people to be here. Immediately. Today," Carson said Monday in Nevada.
Earlier on Monday, Obama held a news conference in Turkey from the G20 Summit where he criticized GOP presidential candidates and Republicans for saying the U.S. should not accept Syrian refugees -- especially those who seek to block the Muslim ones -- and defended his ISIS strategy.
Reading off a phone, Carson delivered a statement to reporters saying Obama is wrong that sending more troops into Syria would be a "mistake."
"I think plenty of mistakes have already been made," Carson said. "As president, I would give the mission to the Pentagon and ask them what they need to accomplish the mission. ... The era of arm-chair quarterbacking our military will end in my administration."
But Carson did rebuke the notion of only allowing in Christian refugees.
"Well, of course we don't apply religious tests, but we should apply ideological tests, and I would be very reticent to bring in people who are ideologically opposed to the ideal of America," Carson said.
As for those refugees who are already here, of which there are more than 2,000, he said the U.S. should put them under surveillance.
"I would watch them very carefully, that's for sure. I would certainly stop any further ingress," he said.
Carson's opponent, former Arkansas Gov. Mike Huckabee, also called on Ryan to act to close the nations borders to would-be terrorists.
"Speaker Ryan needs to make it clear that if the President won't stand to protect America from wholesale open borders, then Republicans will," Huckabee said in a statement. "If Ryan will not lead and reject the importation of those fleeing the Middle East without assurances that we can separate refugees from terrorists, then Speaker Ryan needs to step down today and let someone else lead."
Carson also spent much of the news conference trying to demonstrate his bona fides as commander in chief. The retired pediatric neurosurgeon has been under scrutiny on his foreign policy knowledge after a series of eyebrow-raising statements, including claiming that the Chinese were "there" in Syria and an awkward exchange with "Fox News Sunday" host Chris Wallace over the weekend where he dodged a question about who he would call on first to build a coalition with the U.S.
"I think that's the wrong question," Carson said Monday when asked again. "That's silly, to be honest with you. What we really need to be talking about is the overall plan, the overall strategy, and not who's going to call first."
He also defended his China comments, saying Beijing has sold "sophisticated weaponry" to Syria, and said his comments about his advisers being better than Obama's was "tongue in cheek."
"I have advisers who are former military people and advisers who are former CIA people," Carson said.
But he also had no answer to a question about how NATO powers should invoke the Article 5 provision calling for mutual defense after the attack on NATO ally France, taking a long pause before saying: "I'm not sure."
"They have declared war on us," he said. "Article 5 says if you do something that is seen as an attack on any member of NATO, everybody obviously joins him, but the reason that I'm a little hesitant is because I'm not 100% sure that just saying I vote Article 5 will change anything, I think it's really more what we do than what we say that is important," Carson said.
Pressed for specifics on how he would fight ISIS differently than this administration, Carson actually praised the recent U.S.-supported operation to free Sinjar and cut off a key ISIS supply route, and said he would use "every resource available" in the fight including the world banking system.
He also said the U.S. cannot try to avoid all casualties in the war.
"You cannot fight a politically correct war," Carson said. "There will be collateral damage, and if you're intention is to try to find a way not to have any collateral damage, you're not going to be very effective."