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 official purchasing managers' index (PMI) on Sunday showed that factory activity unexpectedly shrank in October to 49.8, below the 50-point mark, which suggests contraction.
It is also marks the third consecutive month of contraction in manufacturing.
The Shanghai Composite closed down 1.7% to 3,325.08, while Hong Kong's Hang Seng closed down 1.2% at 22,370.04.
A private survey, which focuses on smaller to mid-size firms, also showed on Monday that factory activity fell for the eighth month in a row in October, but it was an improvement from the previous month.
The Caixin/Markit manufacturing PMI was up to 48.3, from 47.2 in September.
Angus Nicholson, market analyst at trading firm IG, said manufacturing activity in China should pick up in the last few months of the year.
"The key takeaway from the manufacturing PMIs is that activity in China has stabilised in the wake of significant monetary easing and fiscal spending," he said in a note.
Japan's benchmark Nikkei 225 index ended down 2.1% to 18,683.24 - leading the losses in the region.
Investors will be watching shares of Nissan after it posted a rise of 37.4% in net income for the six months ending in September.
The focus will also be on US data, with key employment figures due on Friday, which could influence the timing of the Federal Reserve's highly anticipated decision to raise interest rates.
In Australia, the benchmark S&P/ASX 200 closed down 1.4% to 5,165.80 ahead of the central bank's decision on interest rates on Tuesday.
The Reserve Bank of Australia (RBA) is expected to keep interest rates at a record low of 2%.
Westpac shares were down 2.5% after it wrapped up Australia's bank reporting season, seeing a 3% rise in its annual cash profit.
The country's four major banks are preparing for their slowest earnings growth since the global financial crisis, due to low interest rates and a cooling economy.
Bucking the trend, South Korea's Kospi index finished up 0.3% to 2,035.24 points.
The Trump administration is standing firm over its ban on refugees from seven countries despite court rulings and mass protests against the move.
[NEXT_CONCEPT]
Asian markets started the week with losses as investor confidence was dented by China's manufacturing data.