A chorus of Republicans said Friday they would turn their attention to overhauling the U.S. tax code after their legislation to repeal and replace major provisions of the Affordable Care Act couldn't draw enough vote to clear the GOP-controlled House.

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House Speaker Paul Ryan (R., Wis.) said he had spoken with President Donald Trump and his top advisers about moving onto the tax plan. "We have more agreement on the need and nature of tax reform," Mr. Ryan told reporters.

From the Oval Office late Friday afternoon, Mr. Trump told reporters "we are going, right now, for tax reform."

Earlier Friday, Treasury Secretary Steven Mnuchin, at an event in Washington sponsored by media company Axios, said the administration's forthcoming plan to overhaul the U.S. tax code would face smoother sailing than the plan to revamp health-insurance markets that unraveled Friday.

"Health care is a very, very complicated issue," Mr. Mnuchin said. Redesigning the tax code "is a lot simpler."

Taxes could be a challenge as well, however, and lawmakers have been talking about rewriting the tax code for years. President George W. Bush advanced a series of tax cuts in 2001 and 2003, but Washington hasn't completed a comprehensive overhaul of the U.S. tax code since 1986.

Previous GOP attempts have barely gotten out of the gate, blocked by competing business interests and the difficult trade-offs needed to lower tax rates without drastically increasing budget deficits.

Republicans will also have to deal with other issues, including preventing a lapse in government funding in April and raising the federal debt limit in the fall.

Republicans have been planning to advance the tax bill through a process known as reconciliation that would allow them to pass it without Democratic votes. Those rules forbid the plan from increasing budget deficits beyond the budget window, typically 10 years.

Congressional leaders had attempted to use the same legislative vehicle to advance the health-care bill without attracting Democratic support. It ultimately left them with a thin margin for error in the face of internal party divisions.

The stalled effort to replace Obamacare creates new challenges for the Trump administration and congressional Republicans, a point Mr. Ryan conceded Friday. GOP leaders had said the health bill would pave the way to a major tax bill. Repealing Obamacare would cut spending on health-care subsidies and cut $1 trillion in taxes.

Mr. Ryan said Friday the health bill's taxes would now remain and that they wouldn't be repealed as part of the broader tax effort. The upshot is that levies on health-care providers and high-income individuals would remain in place for the foreseeable future.

"Defeat on health care is a blow that could make it harder to pass a more ambitious tax plan such as the one proposed by House Republicans," said Andy Laperriere, a policy strategist at research firm Cornerstone Macro LP. He said the odds had gone up that the ultimate product "tinkers around the margins and is a disappointment to the market."

Mr. Mnuchin dismissed any suggestion that, for political expedience, the White House would forsake comprehensive legislation by seeking instead to pass a handful of smaller tax bills.

"We're not cutting this up and doing little pieces at a time," he said Friday.

Some Trump supporters have criticized the administration for pursuing health-care legislation ahead of the tax overhaul, but Mr. Mnuchin said the administration had taken that route because the tax plan wasn't finished.

"We are designing it from scratch," he said. "We would not have been ready to go a month ago on tax reform and now we are."

Rep. Kevin Brady (R., Texas), chairman of the House Ways and Means Committee, has said his panel will move a major tax bill by the end of the spring, but no bill has been released.

As on health care, Republicans already have faced some divisions on taxes, including over how much they should be allowed to add to deficits, and whether they should come up with new sources of revenue to make up for lower tax rates.

The House Republican plan includes a border-adjustment provision that would tax imports and remove taxes from U.S. exports to raise nearly $1 trillion in new revenue, but the provision has met fierce resistance from some Republican lawmakers in the Senate.

Mr. Mnuchin said the administration has reservations about the border-adjustment provision and said he was worried about how currency movements that result from those tax changes could create headwinds for U.S. exporters.

"It is a very complicated issue," he said. "Whatever we do, we want to make sure it's simple and it works."