The vital pipeline has a capacity of at least five million barrels per day.
The 1,200-kilometre pipeline carries crude from Saudi Arabia's main eastern oil fields to the Red Sea port city of Yanbu in the west.
However, Al-Falih stressed that the production and exports of Saudi crude oil and products will continue "without interruption".
"We need to do a thorough investigation to understand what happened, why it happened, and then come up with reasonable responses short of war", U.S. Ambassador to Saudi Arabia John Abizaid said in Riyadh on Tuesday, according to Reuters.
On Monday, a USA official familiar with US intelligence had said Iran was a leading candidate for having carried out attacks on the four tankers but that the United States did not have conclusive proof Tehran was behind them.
Saudi Energy Minister Khalid al-Falih said May 13 that two of the targeted ships were Saudi oil tankers, one of them about to pick up a load for delivery to the United States.
He urged for global efforts to confront all terrorist groups that carry out such acts, including the Houthi militias in Yemen that are allegedly backed by Iran.
In the face of growing worldwide concern, US Secretary of State Mike Pompeo scrapped a planned visit to Moscow and headed to Brussels instead for talks Monday with European officials, as Britain warned of the danger of conflict erupting "by accident" in the Gulf.
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A team of USA military investigators reportedly believes Iran or its proxies orchestrated the bomb attacks on four commercial vessels anchored off the coast of the United Arab Emirates on Sunday, two of them Saudi oil tankers.
Neither Saudi Arabia nor the UAE gave details on the nature of the attacks or accused anyone of responsibility.
The team of United States military investigators discovered large holes in all four of the affected ships and believe they were caused by explosive charges, the Associated Press reports, quoting an unnamed official.
The Saudi stock index, which suffered heavy losses in the last two days, closed up 0.1% after falling as much as 2.1% on the drone attacks.
Trump dismissed the report as "fake news", but said: "If we did that, we'd send a hell of a lot more troops than that".
"We fundamentally do not seek a war with Iran", Secretary of State Mike Pompeo said at a joint news conference with Russian Foreign Minister Sergei Lavrov in Sochi.
Last month, Iran threatened to "close" it if it was prevented from using the waterway. This latest development is ratcheting up tensions in the already troubled region.
The spike in tensions came after Tehran said Wednesday it had stopped respecting limits on its nuclear activities agreed under a 2015 deal that has since been abandoned by Washington.
Fujairah also is about 140 kilometers (85 miles) south of the Strait of Hormuz, the narrow mouth of the Persian Gulf through which a third of all oil at sea is traded.