Davis said the diplomatic rift has not interrupted or curtailed operations at al-Udeid air base, a launching pad for US military efforts in Iraq, Syria and Afghanistan.
Mr Tillerson also said the blockade by Qatar's neighbours was "hindering U.S. military action in the region and the campaign against Isis".
Saudi Arabia and its counterparts severed ties with Qatar earlier this week over its support to terrorism and accused it of destabilising the region because of the country's ties with Iran and its support for groups like the Muslim Brotherhood and Hamas.
"Until today I have not witnessed Qatar give support to terror", Erdogan said in Istanbul, adding that the Arab states "declare foundations established to provide different services as terror organizations".
Meanwhile, the Indian Embassy in Doha has also issued an advisory to expatriate Indians in the Gulf country asking to them to remain updated about the developments and not to believe in rumours.
"The next step is for Qatar to acknowledge these concerns and commit to reexamine its regional policies", its ambassador to the United States Yousef al-Otaiba told the official WAM news agency.
Secretary of State Rex Tillerson is calling on Saudi Arabia, Egypt, the United Arab Emirates and Bahrain to immediately ease their blockade on Qatar.
UK govt a minority on eve of Brexit talks
By talking tough on immigration and trade she thought she would win over some of the almost four million UKIP voters last time. European officials are anxious that the weaker position of the Conservatives make a breakdown in negotiations more likely.
"We are of the view that all parties should resolve their differences through a process of constructive dialogue and peaceful negotiations based on well-established global principles of mutual respect, sovereignty and non-interference in the internal affairs of other countries", it said. Though the president said others, too, were guilty of supporting terrorism, he said he would not name them.
Qatar hired Ashcroft Law Firm for 90 days, agreeing to pay it $2.5 million to help the Persian Gulf nation comply with USA money laundering and counterterrorism financing regulations, and to stress its efforts to combat global terrorism, according to disclosures filed under the Foreign Agents Registration Act.
Qatar does not support terrorist groups, it stressed, pointing out that "our position on countering terrorism is stronger than numerous signatories of the joint statement-a fact that has been conveniently ignored by the authors". Saudi Arabia, Egypt and the UAE also prohibited Qatari flights from using their airspace.
But US officials insisted both men were sending the same message that the countries of the region should not allow their differences to hinder the fight against extremism.
The country has called the allegations "baseless". Those on the list, including the former interior minister, could not be reached for comment. Some are prominent jihadists who have fought in Libya and Syria.
"Qatar has a history of supporting groups that have spanned the spectrum of political expression, from activism to violence", he added.
After an initial deployment of Turkish soldiers at a base in Doha, Turkish warplanes and ships will also be sent, the mass-circulation Hurriyet newspaper said on its website on Friday.
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