Rep. Mike McCaul (R-Tex.) said Sunday that Republicans don't plan to shut down the government in response to President Obama's executive action halting deportations for some 5 million immigrants.

"We are not going to shut the government down, but we are going to shut down the president and his actions as it pertains to granting amnesty to 5 million people," McCaul, who heads the House Homeland Security Committee, said on CBS's "Face the Nation.”

Obama announced executive actions last week instructing officials to target their deportation efforts only at undocumented workers with criminal records.

McCaul said Obama's plan, announced last week during a prime-time address, "poisons the well going into a new Congress." He added that the president's actions are "a bit of a threat to our democracy."

"It circumvented the will of the American people, circumvented the Congress that has authority under the Constitution to make laws," McCaul said, adding that Obama's responsibility "is to faithfully execute the laws, and I believe he has failed in that responsibility."

Obama has noted that every U.S. president since Dwight Eisenhower has used his executive authority to address immigration issues, although Republicans have noted that the moves by Presidents Ronald Reagan and George H.W. Bush came after Congress approved bipartisan legislation.