The senator also said that lawmakers should approve an amendment to the next Department of Homeland Security spending bill that would rescind Obama's executive actions, forcing the president to either veto funding for the department or back down with his immigration plans.
"We've got to demonstrate that the campaign words used on the trail were more than just talk — that we're willing to honor our commitments," Cruz said, referring to the Republican takeover of the Senate after the recent midterm elections.
Obama said last week when announcing his immigration plans during a prime-time address that every U.S. president for the past half-century has taken similar action. Presidents Ronald Reagan and George H.W. Bush used executive actions to grant legal status to about 1.5 million undocumented immigrants.
Cruz disagreed with Obama, saying those presidents were working with Congress and using executive action to "put congressional will into effect."
"As a matter of constitutional law, the claim the president just made there is frivolous," the senator said. "He acknowledged 22 times, over the last five years, he said over and over again that he had no constitutional authority to do this. Suddenly he decided it would be politically beneficial to do so."
Cruz also suggested that Obama's actions set a dangerous precedent.
"If this president can impose his own immigration laws unilaterally, then the next president can impose his own laws, whether it's immigration, whether it's tax, whether it's environmental," Cruz said. "We stop having a constitutional system of checks and balances that's protecting our liberty."
Rep. Xavier Becerra (D-Calif.) defended Obama's actions. "It's not only legal for the president to take executive action, it's common," the congressman said on the show.