"If we have to act, we will do so," he said.
The Supreme Court is expected to issue its ruling next week on whether the law's requirement for all Americans to buy health insurance violates the Constitution. The House has already passed legislation that would repeal the law entirely, but has also passed several bills that attack the law in piecemeal fashion.
Cantor also noted that the House is still expecting to vote on a privileged resolution holding Attorney General Eric Holder in contempt of Congress for failing to release documents related to the controversial "Fast and Furious" program.
"If the Attorney General would do what it is he committed to do, produce the documents, we'll postpone the vote," he said of papers requested in a House investigation of the botched gun-tracking program. "We have not seen any indication of that, he has not done that, and that's why we've announced the vote."
While the Supreme Court ruling and the Holder contempt vote will dominate the week, Cantor said the House is also planning to pass last-second legislation related to federal highway programs, and a bill dealing with keeping student interest rates low. Authorization for federal highway spending expires June 30, and the interest rate on new, federally backed student loans will double on July 1 without congressional action.
Cantor gave no sense of whether House and Senate conferees would be able to reach an agreement by next week.
"We are desirous of seeing a bill done ... to afford more certainty to the folks who are relying on the funding of our nation's transportation program," Cantor said. "Hopefully we'll have a bill to vote on next week."
However, House Transportation and Infrastructure Committee Chairman John Mica (R-Fla.) and Senate Environment and Public Works Committee Chairwoman Barbara Boxer (D-Calif.) indicated that some progress could be made by next week.
"The conferees have moved forward toward a bipartisan, bicameral agreement on a highway reauthorization bill," Mica and Boxer said. "Both House and Senate conferees will continue to work with a goal of completing a package by next week."
On student loans, Cantor said Republicans have offered proposals to resolve this fight, but that Democrats have failed to react to them. House Republicans have passed legislation cutting a preventive health fund to pay for the lower student loan rate, and more recently proposed to pay for it by requiring federal workers to contribute more to their retirement, or through a combination of ways to save the money.
As busy as next week will be, Cantor said the House would still make time for a White House-Congress picnic and a congressional baseball game.
"We will accommodate both the congressional White House picnic on Wednesday night, as well as the congressional baseball game on Thursday evening," Cantor said. "Debate may continue on appropriations amendments after the picnic and during the baseball game, but during those events no votes will take place."