Mr. Manchin (D., W.Va.) was the pivotal vote, and he explained during a radio interview Friday his reluctance regarding the U.S. going first. His comments don’t officially kill the international tax changes. The White House has said talks will continue, though President Biden on Friday urged adoption of a narrower package focused on prescription drugs and healthcare.

Mr. Manchin has shifted his position at times during a long stretch of negotiation with Democratic leaders. As Democrats run out of time before November’s midterm elections, Mr. Manchin’s statements make it much less likely that the U.S. will adopt the 15% rate soon.

Now the administration must try to urge other countries to go first and hope that momentum, pressure and the potential for lost revenue can compel a future Congress to act.

Ms. Yellen told reporters Saturday in Nusa Dua, Indonesia, where she attended meetings with finance ministers from the Group of 20 major economies, that she was confident that other countries would proceed with the deal, eventually pulling the U.S. along.

“Whether we go first, second or later, the incentives exist for the United States to join this agreement, and we will push forward with every opportunity that we have,” she said.

A Treasury Department spokesman said the U.S. remains committed to the global minimum tax and will look at every avenue possible to get it done.

“We’re sort of at the moment of truth. A significant economy needs to move forward,” said Manal Corwin, a former Obama administration official who is now at KPMG. “The thing that is most likely to move this ahead would be the EU.”

Legislators and negotiators around the world had been waiting to see what the U.S. Congress would do. Mr. Manchin’s comments and his broader objections to his party’s legislation will now reshape the global discussions, putting pressure on other countries to go first in imposing minimum taxes.

“We’re not going to go down that path overseas right now, because the rest of the countries won’t follow,” Mr. Manchin told the radio host Hoppy Kercheval. “And we’ll put all of our international companies in jeopardy, which harm the American economy. Can’t do that. So we took that off the table.”

Backers of the global tax deal might now have to hope that the Biden administration has more success in persuading foreign governments than it did in swaying Congress. In recent weeks and months, the U.S. has been helping to push the European Union toward adoption of the minimum tax, first working to overcome Poland’s objections and more recently putting pressure on Hungary, the current holdout. The U.S. said this month that it would terminate its tax treaty with Hungary.

The Czech government Tuesday said it would aim to secure unanimous backing for the minimum tax at a meeting of EU finance ministers in early October. The Czech Republic has succeeded France as president of the EU, with responsibility for the second half of the year.

The global tax agreement had appeared to be a signal achievement for Ms. Yellen, who helped salvage negotiations that were flailing in the final months of the Trump administration. But the Biden team has been unable to get Congress to pass the legislative changes needed to implement it.

“Other countries understand the outlines of the U.S. separation of powers more than we think they do,” said Lilian Faulhaber, a Georgetown University law professor who has followed the negotiations. ‘“Every time this happens, it means that countries are a little warier of taking the word of one branch of government for what another branch will do.”

Last October’s deal had two key pieces. One, the minimum tax, would put a 15% floor under corporate tax rates. It was designed to help countries raise revenue and prevent companies from shifting profits into low-tax jurisdictions. It was designed to be optional but with mechanisms that encourage nations to join once a critical mass of countries have implemented the tax. That is the part that Mr. Manchin blocked this past week.

The other piece would let countries with large consumer markets impose their corporate taxes on more income and remove unilateral digital-services taxes aimed at U.S.-based technology companies. That proposal, which would require U.S. congressional approval, is moving even more slowly. International negotiators recently extended their deadline into the middle of 2023, and if Congress is then fully or partially controlled by Republicans, U.S. action on that piece would also be difficult to advance.

Even without the 15% minimum tax, the U.S. is already the only major country that has a minimum tax on its companies. In the 2017 tax law, as part of a series of changes to international tax rules, the Republican Congress and then-President Donald Trump created a 10.5% minimum tax.

The proposed global deal called for two major changes to that system: the rate increase and the country-by-country calculation. It also had a set of rules for calculating the taxes and for coordinating the systems across borders.

“I don’t think this is an end,” said David Kamin, a New York University law professor who was a senior White House tax aide until earlier this year. “And there are routes forward for an international agreement to come into place even as there are certainly routes that it doesn’t.”

The consequences for the U.S. and for U.S.-based companies might take time to emerge, and countries and companies will need to figure out how any new systems imposed by some nations interact with the existing ones.

If many countries enact minimum taxes in line with the international agreement, they might be able to tax U.S. companies and effectively take potential revenue away from the U.S. Instead of the U.S. adopting the tax and using its taxing power and clout to prod other countries to join, the reverse will be true.

Most Democratic lawmakers still back the administration position, which was included in a House-passed bill last year. Republicans reject the idea of raising taxes on U.S. companies’ foreign operations, particularly without assurances that other countries will act, and they warn that U.S. companies could be at a competitive disadvantage.

“The Ways and Means Committee will continue to stand behind the administration as we work to honor our commitments,” said Rep. Richard Neal (D., Mass.), chairman of the tax-writing panel. “A deal of this magnitude will have its supporters and its cynics, but I believe a multilateral solution is the best way to end profit shifting and the proliferation of digital services taxes.”

—Paul Hannon and Andrew Duehren contributed to this article.

Write to Richard Rubin at richard.rubin@wsj.com