Gargash said that while Qatar's rivals were "betting on time", a solution could not be brokered until it abandoned its support for "extremist Islamists".
"The sky, there should be no borders up here", a voice says at the start of the video over a blue sky and clouds.
"We are not interested to look at the books because I know how it has been left behind by one of the airlines that was too keen to relaunch it and failed", he said.
Since then, the travel restrictions appear to have been eased slightly.
Saudi Arabia, the UAE and Bahrain broke off relations with Qatar two weeks ago and have closed land and sea borders and imposed severe restrictions on airspace.
Speaking Monday at the Paris Air Show, Akbar Al Baker told The Associated Press: "People will not forget".
"We are taking the modern high ground", he added. "Their offices at the airport have their signage, their lounge is being built at Hammad International Airport [in Doha] by the contractor, we never ever stopped anybody because we do not - Qatar will always act within its legal framework".
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In a separate interview with Reuters news agency, Baker also said the boycott will not halt his company's growth or plans to accept delivery of new aircraft. A base in Qatar houses the headquarters of United States air power in the Middle East. Some have gone viral on social media, using humor to get their messages across.
"We have had a lot of cancellations, especially to the four countries that did this illegal blockade, but we have found new markets and this is our growth strategy", he said, adding Qatar was not the only country affected by the crisis.
There is $18 billion due to banks that have cut ties with Qatar, most of it short-term deposits, Al Jaida said.
Qatar has denied accusations by its neighbours that it funds terrorism, foments regional instability or has cosied up to their enemy Iran.
Qatar would not negotiate with its neighbors as long as it faced a "blockade", Foreign Minister Sheikh Mohammad bin Abdel-Rahman al-Thani said.
The Qatar Financial Centre, which administers special rules for foreign-owned companies operating in Qatar, said on Monday it has no plans to take any action against Saudi Arabian, Emirati or Bahraini firms in response to their governments' sanctions against Doha.
- Jon Ostrower contributed reporting.
