Qatar has stoutly refuted the allegations.
"So good to see the Saudi Arabia visit with the King and 50 countries already paying off", he tweeted in reference to his trip to Riyadh last month.
Trump's call to Qatari ruler Emir Sheikh Tamim bin Hamad Al-Thani, a key U.S. ally, came as a senior Emirati official told AFP that Gulf Arab states were not seeking regime change in Doha.
Qatar is home to the largest United States military airbase in the Middle East, Al Udeid.
Kuwait is leading efforts to find a mediated solution.
"We were surprised" by Trump's comments, Qatar's Ambassador to the U.S., Meshal bin Hamad Al Thani, told the Daily Beast.
Since the diplomatic row erupted, slogans against and in support of Qatar have been among the top topics discussed on Twitter in Arabic, which is a hugely popular medium of expression in the Arab world, particularly in Saudi Arabia.
Tillerson also said the blockade by Qatar's neighbors was "hindering US military action in the region, and the campaign against ISIS", using an acronym for the Islamic State group.
The diplomatic rift between the tiny gas-rich nation and its neighbours threatens to expose the murkier side of British and USA relations with the Gulf states.
The United States, France and Russian Federation have called for dialogue, while Turkey has defended Qatar and said it would further "develop" ties with Doha.
"This ruling applies to all airlines flying into the UAE", the spokesman said in the statement.
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Those breaking ties with Qatar are the UAE, Saudi Arabia, Egypt, Bahrain, the Maldives, Mauritania and Libya's eastern-based government.
The multibillion-dollar preparations to host the 2022 tournament, which involve building nine stadiums and huge infrastructure, is put into perspective by local reports that Qataris are so anxious about the blockade that they are storming the supermarkets and leaving shelves empty of food. But talks were underway to ensure supplies.
Looking to spend quality time with her family in Goa, Sheila Moniz, who works in Bahrain, booked her ticket on Qatar Airways from Bahrain to Goa via Doha for the end of June, well in advance, even factoring the holiday rush and the Eid vacation.
Note: The Qatar Airways call centre numbers in both Abu Dhabi (02 621 0007) and Dubai (04 229 2229) are clearly overloaded and so appear disconnected when calling. Qatar is a major investment destination for UK, US and French energy firms, who have ploughed billions of dollars into gas export facilities which, in turn, play an important role in European and American energy markets.
Among large banks, Doha Bank and Qatar Islamic Bank (QIB) are the most exposed to GCC deposits, with QIB obtaining a quarter of its deposits from the GCC, said Olivier Panis, analyst at Moody's Investors Service.
"We hope that cooler heads will prevail, that wiser heads will prevail and we will not get to that", he said.
Qatar's stock index was down 1.15 percent after plummeting 8.7 percent over the last two days.
Despite the tough rhetoric, USA military officials said that planes continued to take off from Al Udeid in support of US military operations in Iraq, Syria and Afghanistan. Oil briefly rallied before selling off amid speculation that Opec's plan to freeze production and raise benchmark prices could be jeopardised. Qatar's emir has said such a view is a big mistake.
Qatar's Foreign Minister Sheikh Mohammed Bin Abdulrahman al-Than told CNN, "the entire crisis is being based on misinformation". He said his country was "protecting the world from potential terrorists".
Saudi Foreign Minister Adel al-Jubeir said Gulf countries still saw Qatar as a "brother state".
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