"We are not ready to surrender, and will never be ready to surrender the independence of our foreign policy", he told reporters later, adding: "No one will break us".
The president has followed the lead of Saudi Arabia and the UAE, publically backing them in their media war with Qatar.
Qatar acknowledges its links to these groups but denies that it supports terrorism.
"We had a decision to make: do we take the easy road or do we finally take a hard but necessary action?"
"President Trump is right to support the Arab-led effort to change Qatar's behavior once and for all".
Al-Jazeera offices have been shut down by authorities in Saudi Arabia and Jordan.
Qatar's neighbours have given its citizens 14 days to leave, banned Qatari flights from their airspace and closed its only land border.
The deal to deploy Turkish troops in Qatar, aimed at improving the country's army and boosting military co-operation, was signed in April 2016 in Doha.
"The format of the Cooperation Council for the Arab States of the Gulf is the most suitable format for a dialogue of the kind", Qatar's foreign minister said.
"We appeal to the U.S. administration to rely on their own sources and not on countries with political agendas", Meshal Hamad Al Thani said.
The Trump administration tried to defuse the deepening diplomatic rift over Qatar Friday - a situation that it appeared to take credit for earlier in the week.
It wasn't immediately clear how Trump's sharp condemnation might affect US cooperation with Qatar, which hosts some 10,000 USA troops and a major USA air base that serves as a staging ground for operations in Syria, Iraq and Afghanistan.
Only an hour later, Mr Tillerson sat in the front row in the Rose Garden as Mr Trump enthusiastically embraced the move by Saudi Arabia, Egypt and others to punish Qatar. "He just said what he was concerned about", the official said.
He said there are multiple issues at play.
Doha has forged its own alliances in the region for years, often diverging from GCC policies and taking in leaders of the Muslim Brotherhood, the Palestinian Hamas and members of the Afghan Taliban. Speaking in a joint press Conference with the German foreign minister, Sigmar Gabriel, on Wednesday, the Saudi Foreign Minister, Adel al-Jubeir, called Qatar a "brother state" and provided assurances that the crisis could be addressed within the region without outside interference.