"This is nothing new", Spicer said.
"I don't see President Trump backing down", he said.
During a round of Sunday show interviews, Trump's aides stressed that just a small portion of travelers had been affected by the order, which temporarily bars the citizens of seven majority Muslim nations from entering the country.
Priebus also defended the order's implementation, saying there shouldn't have been a "grace period" put on the order.
"So is he confused, or you confused?" the reporter asked.
The Department of Homeland Security said the order also barred green card holders from those countries from re-entering the US. They were the best of us. This includes refugees from its civil war, which has forced 4.8 million people to flee the country since 2011, according to the UN.
"We don't want people that are traveling back and forth to one of these seven countries that harbor terrorists to be traveling freely back and forth between the United States and those countries", he said. "H$3 e's aiming at USA citizen kids who have an undocumented parent, and depending how broad the reach of his order, he could deport kids who have received reduced lunches in school".
The government in Iraq, which is allied with Washington in the battle against ultra-hardline Islamist group Islamic State and hosts over 5000 USA troops, also did not comment on the executive order.
The US must be vigilant during the visa-issuance process to ensure that those approved for admission do not intend to harm Americans and that they have no ties to terrorism, it said.
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Trump said Saturday the order is "not a Muslim ban".
"The overwhelming consensus in the economics academic literature is that immigrants add more to the economy than they take, they create more jobs for Americans, and they are a net benefit to the American economy", said Alex Nowrasteh, an immigration policy analyst at the CATO Institute, a libertarian think tank.
To refresh, Trump issued an executive order on January 27 barring citizens of Iraq, Syria, Iran, Yemen, Somalia, Sudan and Libya from entering the United States for 90 days.
The president on Saturday denied the executive order was a Muslim ban.
The Trump administration sought to clarify the president's controversial action as reports Saturday suggested confusion as officials moved to implement the order.
People in transit at the weekend were detained leading to chaos at airports and widespread protests.
Meanwhile, large crowds of protesters turned out at airports around the country to denounce the new President's ban, The New York times reported.
Joined earlier in the day at the White House by British Prime Minister Theresa May, Trump reaffirmed the United States' "special relationship" with Great Britain. They tell the Muslim world that the United States has friends and interests elsewhere; fearless Iraqis and others from the targeted nations who risked their lives to assist US counterterror efforts are now being told to go fend for themselves.
The other plaintiff, Haider Sameer Abdulkhaleq Alshawi, 33, is the husband of a woman who worked for a USA security contractor from 2006 to 2007 as an accountant, the lawsuit said.
Trump had promised during his campaign what he called "extreme vetting" of immigrants and refugees to try to prevent terror attacks.