But the bigger objective may well be to silence figures such as Qaradawi, who ranks No. 19 on the Gulf Arabs' "terrorist" list but surpasses all others in his ability to sway Muslim opinion, according to former and current US officials and Middle East analysts.
The Trump Administration is currently seeking to serve as mediator in a dispute that officials acknowledge is less about support for groups such as al-Qaeda and Isis (Islamic State) than about Qatar's friendly ties with Iran and support for the kind of activist Islam embodied by the Muslim Brotherhood movement.
To many in the Gulf region, no one better exemplifies the problem than Qaradawi, a man whose beatific smile and folksy speaking style belie a history of defending suicide bombings in Israel and condoning violence against US troops in Iraq, current and former US officials say.
US officials also have criticised Qatar for being slow to shut down financial networks used by Islamist militants, though analysts say the gas-rich kingdom is hardly alone in its support for groups with extremist views.
The Egyptian-born imam has long been regarded as an agitator by Saudi Arabia, the United Arab Emirates, Bahrain and Egypt, the four countries that severed ties with Doha earlier this month.
For decades, Qaradawi's sermons and popular TV show Sharia and Life have been beamed into tens of millions of Muslim households, infused at times with harsh criticism of the Gulf monarchies as well as support for the Hamas organisation and the Muslim Brotherhood, which all four countries have labelled as a terrorist organisation. The cleric has long been regarded as an intellectual leader of the Brotherhood, which won control of the Egyptian Government in a democratic election in 2012 and is regarded by Qatar as moderate.
"Qaradawi is one of the most public figureheads of the radical wing of the Muslim Brotherhood," said Matthew Levitt, a former counter-terrorism official at the FBI and Treasury Department who closely tracked terrorist financial networks in the Middle East. "From the perspective of those who put together the list, this signals a desire for Qatar to take action - not just against terrorist financial networks but against those who support Islamic extremism more broadly."
Qatar has defended its most famous cleric while rejecting as "baseless" the allegation that it harbours extremists. Qatari officials in interviews noted that the 59 alleged terrorist supporters include several figures regarded as political or military opponents of the four countries, such as former members of Egypt's ousted Muslim Brotherhood government. Others, officials said, were foreigners with tenuous ties to the kingdom, and still others are believed to be dead.
More galling to the Qataris was the absence of any mention of support for extremists by the countries levelling the accusations. Saudi Arabia, in particular, has a long history of promoting its own rigidly austere, Wahabist version of Islam, one that has been embraced by extremist movements around the world.
"Our position on countering terrorism is stronger than many of the signatories of the joint statement," Qatar's Foreign Ministry said.
The International Union Of Muslim Scholars, the Qatar-based theological organisation that was founded by Qaradawi and often speaks on his behalf, specifically rejected the inclusion of Qaradawi on the terrorist list, calling the accusations against him "without evidence or proof".
For some of the other names on the list, there is less dispute. Even Qatari officials privately acknowledge that illegal fundraising has occurred. They point to a few prosecutions but say a lack conclusive evidence has hampered efforts to obtain convictions.
At least six names on the list are currently under US terrorist sanctions, including Abd al-Rahman al-Nu'aymi, a Qatari national and founding member of the Qatar-based Eid Charity. Nu'aymi was accused by US officials in 2013 of being a "terrorist financier and facilitator" who supplied money to al-Qaeda-linked groups in Syria, Iraq and Yemen.
A Qatari government official, insisting on anonymity to discuss the diplomatically sensitive charges, noted that his government had frozen the assets of several Qatari nationals on the list and restricted their foreign travel, even of the individuals who had been acquitted by Qatari courts of aiding terrorist causes.
But the official rejected accusations that some of the Qatari organisations on the list illegally raised money. He noted that three of the groups work in partnership with United Nations relief agencies, and another, the Qatar Volunteer Centre, is a tiny operation run by a mother of six who raises about US$2700 a month, partly through bake sales, the official said.
"She sells cupcakes," he said. "She has no relation to terrorism."
The dispute puts the White House in an increasingly difficult spot as arbiter between Gulf neighbours who, despite the increasingly bitter fraternal rivalry, are all strategic allies of the United States.
Washington's relationship with Qatar has itself been notoriously complex as the two countries cooperate closely on military matters, even as they bicker over which individuals and groups they regard as terrorists. Qatar is home to the US Central Command and serves as a launchpad for airstrikes against Isis.
In recent weeks US President Donald Trump has appeared to tilt toward the Saudis, siding with Riyadh in Twitter postings and remarks criticising Qatar's record on terrorism, even as his aides publicly praised the kingdom for recent steps to crack down on terrorist financial networks.
But the concerns about Qatar's role as a financial and ideological base for extremist groups have been shared by Democratic and Republican administrations for at least two decades. Leaked diplomatic cables from the Obama Administration show US officials lodging repeated complaints about Qatari organisations and charities, including Qaradawi himself.
It wasn't so long ago that Qatar's neighbours also expressed admiration toward the preacher who some dubbed the "global mufti".
In Saudi Arabia, Qaradawi was honoured in 1994 with a prestigious award: the King Faisal International Prize, recognising exceptional achievements to humanity in religious scholarship or the secular sciences.
And in 2000, in neighbouring Dubai, he was given an international award as the "Islamic Personality of the Year".
Qaradawi remains popular in many parts of the Gulf, where he is remembered as the spiritual "Dear Abby" who dispensed advice on Sharia and Life - his weekly show on Al Jazeera - on topics ranging from the Arab Spring to female masturbation.
Millions of Muslims worldwide still regard him as a pre-eminent Sunni Muslim scholar and the unofficial intellectual leader of the Muslim Brotherhood.
That's why Qatari officials tend to regard the accusations against Qaradawi as purely political - a reflection, they say, of the cleric's criticism of Saudi Arabia's Wahhabi strain of Islam as well as domestic worries about his popularity. Qatari officials and many US analysts point to Qaradawi's relatively moderate positions on issues such as democracy and interfaith harmony and strong condemnation of Isis, both the organisation and its brutal acts.
But Qaradawi's more incendiary statements have drawn condemnations from Western governments as well as Qatar's Gulf rivals. In 2012, France joined Britain and the United States in refusing entry to Qaradawi, in part because of his statements defending the use of suicide bombers to attack Israelis.
In the past, he has condoned Iraqi resistance against US troops, and has suggested that the murder of six million Jews by Nazi Germany was "divine punishment" for historical transgressions. He has repeatedly called for the destruction of Israel, including the killing of civilians.
"Qaradawi has used his TV programme to promote a fatwa encouraging suicide bombers, as well as to defend the killing of American soldiers in Iraq as a 'religious obligation,'" Yousef al-Otaiba, the UAE's ambassador to Washington, wrote in an essay published last week in the Wall Street Journal.
Whatever sway he does possess is undoubtedly fading. At 91, Qaradawi has retired from television, and his sermons and public pronouncements are less frequent. Yet, his status as Qatar's most venerated imam appears secure.
