President Trump has also dismissed the country's top lawyer, Acting Attorney General Sally Yates after she directed the Justice Department not to defend his Executive Order on this issue.
Families separated from loved ones by the ban share their stories.
Samuel Jacob was born in Syria but lives in Pittsburgh, US. He recently got married but cannot be with his wife.
"I am an instructor in heart and lung transplant at University of Pittsburgh Medical Center.
"Although I was born in Syria, I carry dual US and British citizenship.
"My wife is a dermatologist, is currently living in the south-western city of Sweida in Syria.  She holds a Syrian passport so we are affected by this ban.
"We got married last October in Lebanon and although we keep in touch via Skype, she is currently unable to be with me.
"A lot of money has been spent on lawyers' fees, translation and paperwork but I want to be reunited with my wife.
"Also, I'm nervous about leaving the United States in case I'm not able to get back in again.
"I want President Trump to let me know when my wife can join me."
Helen Honarpisheh is an Iranian citizen, but a permanent resident of the US, currently living in Durham, North Carolina.
"I am a physician and work at a hospital in North Carolina.
"My elderly parents live in Shiraz, Iran and were hoping to visit me. They were last here in 2011.
"If you live in Iran, you either have to travel to Dubai or Turkey to get a visa to visit the US.
"My parents went to Dubai and spent a lot of time and money on their visa application.
"They purchased their tickets and had everything arranged. But now they are banned from coming to the US to visit their daughter.
"I have studied for years, worked hard day and night and paid taxes in this country.
"I believe I am entitled to the very basic human right of being able to see my loved ones.
"This is very demoralizing. It's like I am trapped in a big prison."
The Trump administration is standing firm over its ban on refugees from seven countries despite court rulings and mass protests against the move.