Bush conceded that screening refugees to determine whether they are Christian presents a "national security challenge" to the United States and European countries, but he added that "we do that a lot" already as the country screens potential refugees for relocation.
Bush's views on refugees differ from other GOP presidential contenders who have called for an outright ban on accepting any Middle Eastern refugees. Former Arkansas governor Mike Huckabee has been saying for weeks that the United States should not accept any more refugees from the region. On Friday, former neurosurgeon Ben Carson urged Congress to stop the Obama administration from allowing in more refugees from the regions.
The terrorist attacks in Paris on Friday are "a warning for our country that this threat is not going to go away," Bush added. "This is a threat against Western civilization, and we need to lead. The United States has pulled back, and when we pull back, voids are filled and they’re filled right now by Islamic terrorism that threatens our country.”
Just a few hours after the series of attacks began Friday, Bush told radio host Hugh Hewitt that the clash with the Islamic State and other militant groups targeting Westerners "is the war of our time."
And on Sunday, Bush told NBC's "Meet the Press" that the United States should take formal steps to combat the Islamic State, which is also known as ISIS and ISIL.
"We should declare war and harness all of the power that the United States can bring to bear, both diplomatic and military, of course, to be able to take out ISIS. We have the capabilities of doing this; we just haven't shown the will," he said.
Earlier on CNN, moderator Jake Tapper asked whether the United States and NATO should invoke Article 5 of the alliance charter — that an attack on one member country is an attack on all. Bush said President Obama "should convene" other member nations "to discuss that."
"I do think it’s worthy of consideration, for sure. If that’s what the French want, then — our longest, and strongest and most loyal ally over history — then we should certainly consider it.... We need to show complete solidarity with them."
Bush earned the ire of French citizens last month when he suggested that the U.S. Senate calendar is as easy as "the French workweek," which he said lasts only three days. He later apologized for his remarks, admitting that he "did a disservice to the French" — in part because French workers average nearly 39 hours per week.
On Sunday, Bush also accused Obama of failing to develop a clear strategy to counter the Islamic State, repeating lines on Sunday that he uses almost daily on the campaign trail.
"The president has admitted he does not have a strategy as it relates to ISIS. Hillary Clinton last night said that it’s not our fight," Bush said. "It is our fight, and without our leadership to build a coalition to destroy ISIS, it won’t happen."
Bush called for establishing a no-fly zone over Syria, a move the Obama administration has resisted. He also called for establishing "safe havens" for Syrians fighting against the regime of Bashar al-Assad. And he urged Obama to develop a new coalition, made up mostly of Arab countries, "to support one force" to fight Assad and ISIS.
He also criticized the Democratic presidential candidates for not using the term "radical Islam" to describe the violence during their Saturday night debate in Des Moines. But when Tapper asked him whether he thinks that Islam is a peaceful religion, Bush said: "I know what Islamic terrorism is, and that is what we’re fighting with ISIS, al-Qaeda, and that is what our focus should be on."
"This is a fight for Western civilization. We need to be all-in on it, and the simple thing is that these are Islamic terrorists who have co-opted a religion that is peaceful," he added.