“This is not a manufactured crisis,” Ms. Nielsen said. “This is truly an emergency.”

Sarah Huckabee Sanders, the White House press secretary, later accused Democrats of refusing to acknowledge a crisis at the border. She also chastised Senate Republicans who did not support the president’s emergency declaration.

“If you had done what you were elected to do on the front end, then the president would not have to fix the problem on his own through a national emergency,” Ms. Sanders said.

Some Republicans at the House hearing said they supported building a wall along the border.

Representative Daniel Crenshaw, Republican of Texas, acknowledged the concerns about family separations but said physical barriers were crucial to securing the long stretches of land between ports of entry.

Mr. Crenshaw said he now doubted that “we all value the rule of law and value the ability to manage our border effectively” — a view he admitted was cynical.

The Department of Homeland Security has opened three investigations into the Trump administration’s immigration policies, the agency’s acting inspector general, John V. Kelly, said in a separate hearing on Wednesday.

Mr. Kelly said one investigation is looking into the department’s systems for tracking migrant children separated from their families; another is examining whether asylum-seeking families are being separated at the border.

Department investigators were also reviewing whether parents facing deportation were given the option of bringing their children back home with them. In the hearing with Ms. Nielsen, Ms. Rice asked whether every relative facing deportation was, in fact, allowed to do so.