"We know that most of the American people don't like this law," Cantor said on CBS' "This Morning" Friday. The House will "look toward the kind of healthcare people want," which Cantor, R-Va., said is "patient-centered."
The U.S. Supreme Court declared the law constitutional Thursday.
The House first voted to repeal the law in January 2011, soon after Republicans took control of the chamber.
Cantor said that the ongoing debate over the healthcare law is "all about this election and whether this law is going to go forward or not. ... [Presumptive GOP presidential nominee] Mitt Romney will be the one that will, frankly, get the healthcare that most people want back on track."
Cantor told CBS he didn't support Washington "requiring someone to purchase healthcare and then telling someone what kind of healthcare coverage that's going to be, which is exactly what this Obamacare bill is."
In the Democrat-controlled Senate, Republican Leader Mitch McConnell also said Thursday Congress should repeal the law.
Democrats, for the most part, said it was time to focus on other matters.
"Our Supreme Court has spoken. The matter is settled," Senate Majority Leader Harry Reid said on the Senate floor Thursday. "It's time for Republicans to stop fighting yesterday's battles."
The most controversial aspect of the Affordable Care Act is the individual mandate that requires all Americans to buy health insurance if they can afford it or pay a penalty.
Romney said he opposes a mandate on the federal level and vows to repeal the law if elected. However, he implemented a healthcare mandate in the state of Massachusetts while he was governor.