To "protect its national security from the dangers of terrorism and extremism" Riyadh made a decision to "sever diplomatic and consular ties with Qatar, and to close all land, sea and aviation" links, a Saudi official cited by the official Saudi Press Agency said.
Borders have been closed between the countries and air and sea traffic has been suspended between the peninsular nation and a number of other states.
The US believes it is important the Gulf Cooperation Council, which includes Bahrain, Kuwait, Oman, Qatar, Saudi Arabia and the United Arab Emirates, remains intact, Tillerson said, urging the nations to sort out their differences.
Citizens of these countries have been barred entry to Qatar and Qataris have also been barred entry to these four countries.
This move will effectively cut Qatari Airways, one of the key regional long haul carriers, off from the outside world.
Qatar Airways flies to nine cities in Saudi Arabia.
Qatar Airways said on its official website it had suspended all flights to Saudi Arabia. This follows an accelerating dispute, which has likely ended a 36-year union, between the tiny oil-and-gas-rich country and other Gulf states.
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Saudi Arabian Airlines says it is suspending flights to the Qatari capital, Doha.
Another Jerusalem Post report said that on Saturday Qatar had moved to expel members of Hamas to leave the country due to "external pressure".
"Using sanctions in today's integrated world is inefficient, to be condemned and unacceptable", Ghasemi added of Qatar's neighbours closing all land, sea and air links with it.
But Qatar's foreign ministry insisted there was "no legitimate justification" for the decision, and said it was a "violation of its sovereignty".
Robert Gates, the former United States defence secretary was quoted by Financial Times as saying Tuesday that the USA should demand that Doha should be more aggressive in shutting down terror financing, or risk a downgrading of the US's military presence in the country.
On May 27, Qatar's ruling emir, Tamim bin Hamad Al Thani, called Iranian President Hasan Rouhani to congratulate him on his re-election.
As for the USA response, both Secretary of State Rex Tillerson and Defense Secretary James Mattis say they don't think the decision will affect the fight against terrorism.
Qatar has denounced its diplomatic isolation as based on lies about it supporting militants. The chief worry among them is the Muslim Brotherhood, a Sunni Islamist political group outlawed by both Saudi Arabia and the UAE as it challenges the nations' hereditary rule.