"We are not ready to surrender, and will never be ready to surrender the independence of our foreign policy", he said.
Mr. Erdogan has said isolating Qatar would not resolve any problems.
Sheikh Mohammed also said Qatar's ruling emir will not leave the country while it is "in blockade" to attend a meeting with US President Donald Trump at the White House to end the crisis.
Saudi Arabia, the United Arab Emirates, Egypt and Bahrain lead a string of countries that this week cut ties with Qatar over what they say is the emirate's financing of extremist groups and its ties to Iran, Saudi Arabia's regional arch-rival. In backing the Saudis, the president offered unconditional support for a country that has fostered the spread of Islamist extremism across the world and that has supplied numerous foot soldiers for the Islamic State and al-Qaida - not to mention most of the 9/11 hijackers.
The move saw Qatar's only land border closed and air and sea links cut by the four Arab countries. Davis said the diplomatic rift has not interrupted or curtailed operations at al-Udeid air base, a launching pad for USA military efforts in Iraq, Syria and Afghanistan.
The foreign minister of Oman met Sheikh Sabah Al-Ahmad Al-Jaber Al-Sabah, Kuwait's ruler, for talks. Qatar denies the allegation. This is partly due to the significance of the al-Udeid air base, from which almost all coalition airstrikes against Islamic State are being conducted. The regional headquarters of U.S. central command is hosted by Qatar at the Al Udeid airbase.
Anxious residents have responded to the crisis by emptying grocery stores in the capital of Doha, and Saudi Arabia has blocked trucks carrying food from entering the country across its only land border.
For years, Gulf states and Egypt have resented Qatar's ties to the moderate Muslim Brotherhood movement along with its sponsorship of the pan-Arab television network al-Jazeera.
But despite his defiance, the foreign minister was adamant the crisis would not evolve into a military conflict.
Turkey has brought forward a troop deployment to Qatar and pledged to provide food and water supplies to its Arab ally, which hosts a Turkish military base. The agreement did not detail a timeframe or the number of troops.
As the crisis has expanded, Doha, the world's largest exporter of natural gas, has called on its strategic allies and important partners around the world to intervene.
"We don't know, all these measures, why they are being taken", he said.
Sheikh Mohammed is expected in Moscow Saturday to meet Foreign Minister Sergei Lavrov, and officials said Friday he also spoke with Tillerson by telephone.
"Now we need to check how the neutral venues will be impacted and we will have to go back to the countries that have chosen Doha as a neutral venue to come up with alternatives". Qatar's neighbors have also criticized Qatar for allegedly supporting groups like the Muslim Brotherhood and Hamas.
Qatari investigators have said that a cyber attack which caused the publication of fake news last month that helped trigger a crisis with Gulf neighbours was prepared in April.
Gargash said "not using the formidable media ownership in promoting an extremist agenda" was a condition for negotiations.
On Friday Saudi Arabia, the UAE, Egypt and Bahrain designated as terrorists five Libyans including Tripoli Grand Mufti Sadiq al-Ghariani, an influential figure for anti-Haftar militias in western Libya.