Republicans thought they’d found a solution Tuesday to the separation of illegal immigrant families on the border, announcing plans to let the families stay together in Homeland Security detention facilities while they await deportation.

But Democrats quickly shot down any thought of signing on, saying they’re in no rush to help Mr. Trump clean up the mess they say he’s made.

“There’s no need for legislation,” said Senate Minority Leader Charles E. Schumer, New York Democrat. “Mr. President, you started it, you can stop it, plain and simple.”

His resistance suggests Republicans will struggle to find an honorable exit from an issue that’s playing poorly for them amid wall-to-wall coverage from a hostile press corps, denunciations from religious leaders, and official protests from foreign countries.

It also means that unless Mr. Trump blinks, there’s little chance of relief for the nearly 70 children who are being separated each day.

The president didn’t sound like a man on the verge of changing his mind, delivering a fiery speech to a small business group Tuesday defending his administration’s handling of the policy and saying it’s up to Congress to fix things.

“When you prosecute the parents for coming in illegally, which should happen, you have to take the children away,” he said. “Now, we don’t have to prosecute them. But then we’re not prosecuting them for coming in illegally. That’s not good.”

He blamed Mexico for failing to stop the families as they traverse that country, Democrats for being unwilling to find solutions to the new surge of illegal immigration, and the immigrants themselves for taking advantage of “loopholes.”

And he accused reporters of being complicit with the cartels that control the human smuggling traffic into the U.S., saying the press is distorting the policies.

“They’re fake,” he said. “They are helping these smugglers and these traffickers like nobody would believe.”

Homeland Security officials said that from May 5 through June 9 they arrested 4,538 people who sneaked into the U.S. traveling as families and who faced separation. That included 2,206 adults who were prosecuted and sent to jail to await their proceedings, leaving 2,342 juveniles as Unaccompanied Alien Children who had to be stuck in government-run dorms.

Usually the adults serve a few days in jail, plead guilty to illegal entry charges, are sentenced to the time they already served, and are released back into the immigration system.

Under the law, though, their children have been placed in the custody of the federal Health and Human Services Department, complicating efforts to reunite the families. Officials couldn’t say how many of the 2,342 children had been united as of this week.

Audio of children sobbing for their parents and photos of families being questioned by Border Patrol agents have swept across the Internet this week, bringing the issue home for many Americans who say they’re troubled by what they saw.

The same images were captured under Presidents Obama and George W. Bush. Indeed, families were separated during both administrations. But the attention is much higher on Mr. Trump, as are the numbers of children being separated.

The Trump administration pins blame for the situation on the Flores settlement, which imposed strict conditions on the care of illegal immigrant children, and on a federal law, the Trafficking Victims Protection and Reauthorization Act.

Under those policies, children nabbed at the border with their parents generally cannot be held in detention longer than 20 days. And when the children are released, their parents are generally released as well. They disappear into the shadows, rarely showing up for their deportation hearings.

That’s created an incentive for adults to show up at the border with children, hoping to take advantage of the leniency.

Mr. Trump’s zero tolerance policy was meant to change those incentives, using the criminal code to punish illegal immigrants. But a consequence was that when parents were charged and taken to jails, their children couldn’t follow.

Senate Judiciary Committee Chairman Charles E. Grassley said he’s pushing a bill that would repeal the Flores settlement, which would allow the families to be detained longer than 20 days together in immigration detention. That would erase the incentive for Mr. Trump to use the criminal-justice system.

He wants to see action in his committee this week on a bill.

House Republicans, meanwhile, proposed a solution that would allow illegal immigrants charged with misdemeanors to wait in immigration facilities rather than jails.

Democrats had their own plans. Sen. Dianne Feinstein, California Democrat, had proposed a ban on family separations — though that bill would have reopened the Flores “loophole” that Mr. Trump and Republicans want to get rid of.

Other Democrats have suggested alternative detention methods other than holding families in facilities.

But those plans were sidetracked by Mr. Schumer, who signaled Democrats will force Mr. Trump to go it alone.

“Anyone who believes this Republican Congress is capable of addressing this issue is kidding themselves,” he said. “The president can end this crisis with the flick of his pen, and he needs to do so now.”

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