Johanns told reporters on Thursday that the ruling specifies that the Senate determines when the upper chamber is in recess, not the president.
"What I support is legal appointments," he said.
The bill, also backed by Sens. Lamar Alexander (R-Tenn.) and John Cornyn (R-Texas), would prohibit the NLRB from taking any action or enforcing any decisions that required the approval of a quorum of members since the appointments were made one year ago.
The NLRB lacked a quorum before the president made the three appointments to the five-member board — the bill would effectively freeze all work the NLRB board had done in the last year.
The bill is not asking the appointees to return any money they have received so far.
"We're asking the President to sit down with us and work on these issues," Johanns said.
But, ultimately, the president will have to "decide what he wants to do."
The measure also would prevent the CFPB from receiving any more funds from the Federal Reserve to do anything that requires a director. The CFPB's funding structure, in which it receives funds from the Fed, has long been a sore spot for Republicans, who argue the agency should receive congressionally appropriated funds, giving lawmakers more control over the bureau. The CFPB does have the power to do some of its work even without a director, but other aspects of its mission, most notably its enforcement of consumer protection laws for nonbank financial institutions, require a director.
The two agencies could not continue their work until they have Senate-confirmed officials in place.
"American democracy was born out of a rejection of the monarchies of Western Europe, anchored by limited government and separation of powers," said Cornyn. "We refuse to stand by as this President arrogantly casts aside our Constitution and defies the will of the American people under the guise of defending them.”
The White House has said it strongly disagrees with the court ruling, but has not said yet if it plans to appeal it. The administration has also argued the ruling does nothing to disrupt the NLRB's work.
The CFPB is facing a separate legal challenge. The president re-nominated Cordray for the position earlier this month, but there are no indications Republicans are any more willing to consider the pick now than they were when they blocked his nomination over a year ago.
Alexander said his problem isn't with Cordray but, instead, with the agency's "lack of accountability."
—This story was updated at 3:30 p.m.