The current closure of a quarter of federal agencies is the second-longest in history, leaving hundreds of thousands of government workers unpaid.
Exasperated Democrats called Trump's behaviour a "temper tantrum" and said the meeting broke down when they refused to commit to funding his proposed southern border wall.
At present, the government has been partially shut down for 18 days as Trump struggles to reach a deal with congressional Democrat leaders to fund the wall.
When she said "no", Schumer said, Trump got up and said: "Then we have nothing to discuss". "Again, we saw a temper tantrum because he couldn't get his way, and he just walked out of the meeting". "Nancy said, NO. I said bye-bye, nothing else works!"
"It's cold out here, and the temperature wasn't much warmer in the Situation Room", she said outside the White House.
MICHAEL DWYER APGovernment workers and their supporters protest the shutdown in Boston
"This should have been done by all of the presidents that preceded me", Trump said.
Kevin McCarthy, Republican leader in the House of Representatives, said he found the Democrats' behaviour "embarrassing".
"Unfortunately, two things caught in the middle are securing our borders, and the men and women in federal government", he added. "But, all Americans are hurt by uncontrolled illegal migration", he said.
As for congressional Republicans, the speech was a demonstration that Mr Trump is going to use every arrow in his presidential quiver to get his wall. A deal along those lines was almost achieved previously but collapsed. Observing that some people have suggested that a barrier is immoral, Trump asked why do wealthy politicians build walls, fences and gates around their homes.
He said: "He asked Speaker Pelosi, 'Will you agree to my wall?' She said no. We need border security". Large metal detectors also could be installed to scan trucks and buses for illegal activities.
Trump demands USA border wall, sidesteps declaring emergency
Senator Doug Jones of Alabama said, "We need to dial back the rhetoric and not use fear". The president said that all Americans are hurt by uncontrolled illegal immigration.
Trump offered no new solutions at a nationwide televised address on Tuesday evening, but instead insisted that a wall was necessary to stem the flow of illegal immigration.
But the president would be accused of usurping Congress's constitutional power of the purse, and the move would be bogged down in a quagmire of legal challenges.
Mrs Pelosi and Mr Schumer have vowed they will not give Trump any money to build a wall.
Mr Trump's approval rating continues to hover around 40 percent.
He concluded: "The symbol of America should be the Statue of Liberty, not a 30-foot wall".
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That shutdown has left about 800,000 government workers furloughed or working without pay, and is affecting national parks, airline security screening, housing and food aid, and economic data.
Union officials said some TSA officers have already quit because of the shutdown and many are considering quitting.
She blamed Trump's "obsession with forcing American taxpayers to waste billions of dollars on an expensive and ineffective wall - a wall he always promised Mexico would pay for!"
Blocking a bill to fund the Treasury Department, for example, might delay Americans' tax refunds.
But there was no indication that the Senate, controlled by Trump's fellow Republicans, would allow a vote on the bill.
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For example, an investigation in 2013 found that a staff member had made "inappropriate, sexual statements " about four clients. It is also unclear if staff members at the Hacienda de Los Angeles facility were unaware of the pregnancy until the birth .