House Republicans wrapped up impeachment hearings against Alejandro N. Mayorkas, the homeland security secretary, on Thursday, a little more than a week after the proceedings began, racing to charge him with failing to enforce the nation’s immigration laws.

The Republicans are moving ahead without evidence that Mr. Mayorkas has committed high crimes or misdemeanors, the constitutional standard for impeachment, effectively seeking to remove him for immigration policies they argue have imperiled the American public.

They plan to bring charges against Mr. Mayorkas as early as the end of this month, without having featured testimony from him or any other witness from the Biden administration to publicly answer for his conduct, or a single constitutional expert to support their argument that he is guilty of impeachable offenses.

Republicans on the House Homeland Security Committee invited the secretary to testify at Thursday’s hearing, but when Mr. Mayorkas asked for a different date, they told him instead to submit a written statement by Jan. 28.

“The truth is that Secretary Mayorkas has disregarded court orders, laws passed by Congress and has lied to the American people,” Representative Mark E. Green, Republican of Tennessee and the chairman of the panel, said during the hearing. “Who wants a secretary that can just disregard the fundamental pillars of the Constitution? We cannot tolerate that, whether they are a Republican or Democrat.”

The Republican push to impeach Mr. Mayorkas comes as right-wing lawmakers attack the Biden administration’s immigration policies and threaten to vote against funding the government unless officials crack down on migration at the U.S. border with Mexico. G.O.P. leaders have insisted Mr. Mayorkas is personally responsible for failing to end a surge of illegal migration and drug trafficking at the border that even Democrats acknowledge has worsened in recent years.