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8 year oldSharia is a legal code based on the Koran and other Islamic scriptures.
Mr Gingrich's comments follow an attack in the French city of Nice, which has killed at least 84 people.
His comments echo the sentiment held by likely Republican presidential nominee Donald Trump, who has said Muslims should be banned from entering the US.
"Western civilisation is in a war. We should frankly test every person here who is of a Muslim background and if they believe in Sharia they should be deported," Mr Gingrich told Fox News.
"Sharia is incompatible with western civilisation. Modern Muslims who have given up Sharia, glad to have them as citizens. Perfectly happy to have them next door," he added.
Hours earlier, a driver, identified by French media as Mohamed Lahouaiej Bouhlel, rammed a lorry through a crowd of people celebrating Bastille Day before he was shot dead by police.
Witnesses say the 31-year-old French Tunisian swerved the vehicle in an attempt to hit more people.
Police reportedly discovered guns and grenades inside the lorry.
No group has claimed responsibility but French President Francois Hollande described the tragedy as a terror attack.
Mr Gingrich, who was in the frame as Mr Trump's potential running mate beforeIndiana Governor Mike Pence was asked, also suggested monitoring US mosques and jailing anyone who visits websites favouring a terror group.
"Anybody who goes on a website favouring Isis, or al-Qaeda, or other terrorist groups, that should be a felony, and they should go to jail," he added.
The Council on American-Islamic Relations condemned the attack in Nice as well as the former House Speaker's comments.
CAIR National Executive Director Nihad Awaid said Mr Gingrich's suggestion that American Muslims be subjected to an "Inquisition-style religious test" plays into the hands of terror recruiters and "betrays American values".
Mr Trump, who will be confirmed as the Republican presidential nominee next week in Cleveland, sparked international outrage last year after calling on the US to ban all Muslims from entering the country.
The proposal was roundly condemned by Democrats and many Republicans, including his expected running mate Mr Pence.
He later suggested the ban apply only to people coming from countries with a terror history against the US.
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