A similar statement by the United Arab Emirates, with a possible 15-year penalty, was issued Wednesday, prohibiting criticism of that government or sympathy toward Qatar, "whether it be through the means of social media, or any type of written, visual or verbal form."
The measures, in countries where civil rights are already sharply restricted, came as Al Jazeera, the state-funded Qatari media organization, said its website and digital platforms were "under cyberattack" and facing "systematic and continual hacking attempts."
President Trump met Thursday at the White House with Defense Secretary Jim Mattis and Secretary of State Rex Tillerson to discuss the situation and reiterated his proposal that Tillerson help mediate the dispute among Arab governments, according to State Department spokeswoman Heather Nauert.
But "the American preference is for the GCC countries to arrive at a solution of their own," Nauert said.
On Monday, three members of the six-nation GCC, or Persian Gulf Cooperation Council — Saudi Arabia, the UAE and Bahrain — severed relations with Qatar, a fourth member. They were joined by Egypt and, since then, several other Arab nations.
In statements, the Saudis and the others have demanded that Qatar distance itself from Iran and stop support for "terrorist groups." Qatar "has to realize its interests are with us, not with another country that conspires against us, wants to dominate and divide us," Bahrain Foreign Minister Khalid bin Ahmad al-Khalifa told Asharq al-Awsat, a Saudi-owned newspaper.
Other countries, including Turkey, have voiced support for Qatar. Kuwait and Oman, within the GCC, have tried to mediate.
Trump's offer of U.S. mediation was first voiced in a telephone call Wednesday with Qatar's emir, Sheikh Tamim Bin Hamad al-Thani, in which the president offered to hold a meeting of regional leaders at the White House.
A White House readout of the call took a markedly different tone from Trump's first comments on the rift, in Twitters postings Monday congratulating Saudi Arabia for taking action against Qatar. Trump also credited his own meetings with the Saudis, the GCC and other Muslim leaders in Riyadh late last month with sparking the action.
Those initial tweets also differed sharply from statements by Tillerson and Mattis, who at the time were visiting Australia and called for Persian Gulf unity. The United States directs its air wars in Syria, Iraq and Afghanistan from a huge air base in Qatar where more than 10,000 U.S. service members are stationed.
On Wednesday, the readout of Trump's call to Qatar said he "reiterated" that a united GCC and its partnership with the United States "are critical to defeating terrorism and promoting regional stability."
In the past, the Treasury Department has said that Saudi Arabia and Qatar, among others in the region, have provided support for Sunni extremist groups. More recently, it has said that governments in the gulf have taken steps to stop money flows, but it has singled out Kuwait, and to a lesser extent Qatar, for failing to stop terrorism financing by individuals in those nations.
A number of regional experts have charged that Trump's lavish praise of Saudi King Salman during his visit to Riyadh effectively encouraged that kingdom to act on long-standing disputes with Qatar over its less restrictive foreign and domestic policies, and what the Saudis charge is biased reporting by Al Jazeera.
Qatar's foreign minister, Sheikh Mohammed bin Abdulrahman al-Thani, speaking to reporters in Doha on Thursday, said his government had not received a list of specific demands from the Saudi-led bloc, but he suggested Qatar was unwilling to modify its policies to satisfy their complaints.
"We are not ready to surrender, and will never be ready to surrender, the independence of our foreign policy," Thani said, according to Al Jazeera.
Kareem Fahim in Istanbul contributed to this report.