He said diplomacy remained a priority, but added that mediation efforts to resolve the dispute had been undermined by the public disclosure of the demands.
"The mediators' ability to shuttle between the parties and try and reach a common ground has been compromised by this leak," he said. "Their success is very dependent on their ability to move but not in the public space."
The demands are apparently aimed at dismantling Qatar's two-decade-old interventionist foreign policy, which has incensed its Arab neighbours.
Gargash said if Qatar fails to comply within the 10-day timeline set out in the ultimatum, it would be isolated, but he did not make clear what more could be done since the four Arab nations have already cut diplomatic relations with Doha and severed most commercial ties.
Kuwait is helping mediate the dispute as is the United States, for which it is a big test since Qatar is home to a base housing the headquarters of its Middle East air power and 11,000 troops.
The countries that imposed the sanctions accuse Qatar of funding terrorism, fomenting regional unrest and drawing too close to their enemy Iran. Qatar rejects those accusations and says it is being punished for straying from its neighbours' backing for authoritarian rulers.
The uncompromising demands leave little prospect for a quick end to the biggest diplomatic crisis for years between Sunni Arab Gulf states.
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