Price's confirmation hearing in the Senate is scheduled for Jan. 18.
An Associated Press story earlier Wednesday noted that Price faces calls for investigation of whether stock trades he made on health-care companies were guided by insider knowledge he had as a result of being a congressman. Trump's transition team has denied any wrongdoing by Price.
At a news conference in New York on Wednesday, Trump did not answer a reporter's question about whether his replacement plan for Obamacare would cover the same number of people — 20 million or so — who gained insurance under the health-care law.
However, Trump did say that "you're going to be very proud of what" will be proposed as a replacement for the Affordable Care Act.
Americans will have "health care that is far less expensive and far better," Trump said at his first post-election news conference, which was held at Trump Tower.
Trump's promise for a quick repeal and replace of the ACA came a day after he told The New York Times of his intention for such a two-step on Obamacare.
But that promise of concurrent repeal and replace, in such a short time frame, is widely considered impossible to meet.
That's because while Republicans, who have 52 votes in the Senate, need only 50 votes to repeal Obamacare, they would need 60 votes to overcome a likely filibuster for any replacement of the landmark health-care law. To reach 60 votes, eight or more Democrats would need to agree to vote for a replacement, which is unlikely given that party's current position opposing the repeal of the ACA.
Meanwhile, a growing number of Republicans have voiced opposition to the idea of repealing the law without having a replacement plan on deck.
Potentially complicating matters further for the GOP was news Tuesday that enrollment in Obamacare plans so far this sign-up season is outpacing the rate seen at the same time last season. And the Council of Economic Advisers said that big price hikes for key Obamacare health plans in 2017 seem to be a one-time correction.
Both developments go against Republican arguments that Obamacare is doomed to failure.
Already, Republicans were leery about committing verbally to promises to keep the same number of people covered under a replacement as have gained insurance from the ACA.
Obamacare has driven down the nation's uninsured rate to record levels in three ways: expanding Medicaid to more poor adults; offering subsidies for low- and moderate-income people to buy individual health plans; and letting adults under age 26 be covered by their parents' plans. The ACA also mandates that nearly all Americans have health coverage or pay a fine.
But Trump said Wednesday that there is an urgent need to get rid of Obamacare.
"Obamacare is a complete and total disaster," Trump said. "It's imploding as we sit."
He said that people who buy Obamacare plans are being strained by rising premiums and sky-high deductibles.
"And '17 is going to be the bad year, it's going to be catastrophic," Trump said.
The president-elect said he and his fellow Republicans were doing Democrats a "tremendous service" by killing the law passed under their leader, President Barack Obama — and suggested the GOP could actually get a better replacement plan passed if they allowed the law to collapse further.