The four countries have issued 13 demands, including the closing of Al Jazeera, Qatar’s influential television channel, and, more broadly, the abandonment of Qatar’s foreign policy, which includes support for a wide variety of Islamist factions.

In London, Qatar’s foreign minister, Sheikh Mohammed bin Abdulrahman al-Thani, accused his country’s foes of “clear aggression” and said Qatar was ready for a lengthy standoff, having developed new supply routes for food, construction materials and other imports. He singled out Saudi Arabia and the Emirates as the main foes of Qatar, accusing them of seeking to make it surrender its sovereignty.

That, he said, “Qatar will never do.”

The uncompromising statements offered little hope for a speedy resolution to a rift that opened up on June 4 with the sudden blockade on Qatar. The crisis has worried many Western countries that are concerned about critical military, business and energy interests, and are fearful that the region is tipping into a dangerous and unpredictable situation.

The confrontation on Wednesday could have grown even more serious. But defying expectations, the ministers gathered in Cairo avoided imposing new sanctions on Qatar, and instead sought to reframe their demands as a series of broad principles about combating extremism and not destabilizing each other’s government.

Confusion over the United States’ stance on the dispute may be exacerbating the problem.

President Trump called leaders on both sides of the ill-tempered dispute on Sunday, but he has left little doubt that he is siding firmly with Qatar’s opponents. American officials say that Mr. Trump sees the crisis as an opportunity to force changes in Qatar’s maverick foreign policy, and to bolster his close alliance with the rulers of Saudi Arabia and Egypt. In Cairo on Wednesday, the ministers thanked Mr. Trump for his “firm stance on extremism and terrorism.”