Speaking at the Asia-Pacific Economic Cooperation summit in Vietnam, the president also warned against mistreating his country, accusing some countries in the region of "product dumping, subsidized goods, currency manipulation and predatory industrial policies."
"We can no longer tolerate these chronic trade abuses, and we will not tolerate them," Trump said.
"From this day forward, we will compete on a fair and equal basis," he added. "We are not going to let the United States be taken advantage of anymore. I am always going to put America first the same way that I expect all of you in this room to put your countries first."
Additionally, Washington will "no longer tolerate the audacious theft of intellectual property, we will confront the destructive practices of forcing businesses to surrender their technologies to the state and forcing them into joint ventures in exchange for market access," he said.
Many U.S. firms operating in China have complained of such policies.
Trump said he did not blame China, the world's second-largest economy, or other countries for "taking advantage of the U.S. on trade," repeating a message he delivered in Beijing the previous day.
Those who play by the rules will be Washington's closest economic partners, Trump said. "Those who do not can be certain that the United States will no longer turn a blind eye to violations, cheating or economic aggression. Those days are over."
"We cannot achieve open markets if we do not ensure fair market access," he said, adding that unfair trade "undermines us all."
The world's largest economy will also sign bilateral trade agreements with Indo-Pacific nations that abide by fair and reciprocal trade, he continued. "What we will no longer do is enter into large agreements that surrender our sovereignty."
Trump pulled his country out of the Trans-Pacific Partnership agreement, pegged as the world's largest trade deal, in January on concerns about American jobs.
The U.S. leader also praised the economic transformation of many Asian nations, including Vietnam, Indonesia, Thailand and the Philippines, saying Washington has been an active partner in Asia-Pacific since the U.S. first won independence.
"We seek friendship and we don't dream of domination."
Trump arrived in Vietnam on Friday as part of a milestone Asia tour that will take him to Manila next.
The APEC summit, taking place in this year, is focused on regional trade, with particularly attention on the TPP.
Since Washington's withdrawal from the landmark accord in January, the pact's 11 remaining members have been discussing how to move ahead with the agreement, which seeks to remove tariffs across certain industries as well as safeguard labor, environmental and intellectual property rights. A framework was expected at this week's APEC summit, but nothing final has been revealed so far.
Trump was widely expected to hold a separate meeting with Russian President Vladimir Putin, who will also be at the APEC summit. But that won't be possible due to scheduling conflicts, the White House said Friday.