President Donald Trump issued a challenge after a federal appeals court unanimously decided to uphold a suspension of his executive order.
Trump's order restricted travel for people from seven majority-Muslim countries.
Last week, US Judge James Robart issued a temporary restraining order, halting enforcement of the president's action.
Following news of the ruling, Trump tweeted, " SEE YOU IN COURT, THE SECURITY OF OUR NATION IS AT STAKE!"
Trump told NBC News that the ruling was a "political decision," echoing comments he made at a law enforcement conference earlier this week.
The president expressed confidence that his administration would ultimately win this legal fight.
Following the unanimous decision by the three-judge panel, Hillary Clinton tweeted "3-0."
The Justice Department has argued on behalf of the White House that the order was a "lawful exercise" of presidential authority.
On Monday, the department argued that the injunction reinstated "procedures that the President determined should be temporarily suspended in the interest of national security."
The court said in a Thursday filing that the government has not been able to point to evidence that a foreign national from any of the countries listed in the executive order has perpetrated a terrorist attack in the US
Trump has previously cited security as the basis of his measure.
The court acknowledged the amount of public interest on both sides of the issue.
"On the one hand, the public has a powerful interest in national security and in the ability of an elected president to enact policies. And on the other, the public also has an interest in free flow of travel, in avoiding separation of families, and in freedom from discrimination," the court said in a Thursday filing.