Trump also said that he expects the bill "to get better" as it moves through the legislative process.
"Most importantly, yes, premiums will be coming down. Yes, deductibles will be coming down," Trump said.
He said passage of the bill is a necessity because of ongoing problems with Obamacare.
"I predicted it, a long time ago, I said, 'It's failing,'" Trump said. "And now, it's obviously dead. Dead!"
Trump, who delayed a planned visit to New York City to celebrate the House passage, lauded the Republicans behind him for winning the vote almost two months after they aborted an earlier vote because of lack of support from some GOP members.
The Republican leadership signed off on a series of amendments to the bill to satisfy concerns about it among conservatives and moderates.
"We have an amazing group of people behind us. They worked so hard, they worked so long," Trump said. "This has brought the Republican Party together."
"They're not even doing it for the party," Trump said. "They're doing it for this country because we suffered with Obamacare."
In particular, Trump cited House Speaker Paul Ryan, R-Wis., joking that he had heard complaints about Ryan earlier in the week when it was not clear that the bill would pass, and also hard calls to oust Ryan as speaker.
But on Thursday after the vote, Trump quipped, "I hear Paul Ryan's a genius."
However, even as Trump spoke, a number of Republican senators were saying they will not use the House bill as the basis for an Obamacare replacement bill, but instead will craft their own legislation.
Bloomberg quoted Alaska Sen. Lisa Murkowski as saying she hopes the Senate will begin the process with "a clean slate."
Watch: Healing moment for House Republicans