Saudi Arabia, Bahrain, Egypt and the United Arab Emirates, which lead a boycott against Qatar, have described Doha’s rejection of their demands as a threat to regional security.
In a statement, the Saudi-led bloc also warned of unspecified new measures.
Last month they cut all ties with Qatar, in effect imposing a land blockade on the tiny emirate.
They demanded that Qatar shut down Al Jazeera TV, reduce ties with Iran and end its alleged support for jihadists.
Earlier this week Qatar rejected the four Gulf states’ ultimatum. It denies the allegations against it.
Secretary of State Rex Tillerson is expected on July 10 to visit Kuwait, which is mediating the Gulf crisis.
In a July 6 statement, the bloc said Qatar’s rejection of their 13 demands “reflects its intention to continue its policy, aimed at destabilizing security in the region”.
Saudi Arabia, Bahrain, Egypt and the United Arab Emirates also threatened new political and economic measures against Qatar, without providing any further details.
Qatari Foreign Minister Sheikh Mohammed bin Abdulrahman al-Thani has described the cutting of ties with his country as “a siege that is a clear aggression and an insult”.
He said earlier this week: “The answer to our disagreement is not blockades and ultimatums, it is dialogue and reason.”
Qatar is dependent on imports to meet the basic needs of its population of 2.7 million.
As Qatar’s only land border is now closed, food is having to be shipped or flown in.