E.U. antitrust regulators order Apple to pay up to $14.5 billion in taxes plus interest to the Irish government after ruling that a special scheme to route profits through Ireland was illegal state aid. (Reuters)

European authorities ruled Tuesday that Apple owes more than $14.5 billion in back taxes after striking a sweetheart deal with Ireland that allowed the tech giant to underpay for more than a decade.

Apple paid a tax rate of 1 percent or even less — 0.0005 percent, in some years — on its European profits, according to the European Commission, which launched an investigation into the company’s international tax strategies in 2014. That arrangement, according to the commission, is illegal because it amounts to an improper incentive under European rules.

The ruling is likely to rattle many U.S. corporate boardrooms where aggressive tax strategies have become standard practice. The European Commission also is investigating tax deals reached by Amazon.com and McDonald’s in Europe and has said it will launch other probes soon. (Amazon chief executive Jeffrey P. Bezos also owns The Washington Post.)

The investigations also create a sore spot for the Obama administration: It has been frustrated by U.S. companies that it says are dodging American taxes, even as it forcefully defends those same companies against European authorities. The Treasury Department has complained that U.S. multinationals have been unfairly targeted by the E.U. and said Tuesday that it was “disappointed” by the ruling.

The more pressing issue for the administration is that if Europe collects more cash from U.S. companies, that will leave less to be taxed by the United States. The dispute could escalate tensions between the United States and Europe over how — and where — global companies pay taxes.

European Union Competition Commissioner Margrethe Vestager at a news conference in Brussels after a decision that Apple must repay up to $14.5 billion in unpaid taxes from its European base in Ireland. (John Thys/AFP/Getty Images)

“Perhaps the [European Commission] decision will spur the U.S. authorities to step up their own efforts to collect more taxes from Apple. Apple may be a tax cheat, but Apple is our tax cheat,” said Steven M. Rosenthal, a fellow at the nonpartisan Tax Policy Center.

The European Commission is seeking much more than many tax experts expected, a record amount that could grow if interest is collected.

But the $14.5 billion in back taxes is just a slice of Apple’s cash stockpile. The firm has more than $200 billion in cash, most of it held overseas where it cannot be taxed by the United States. It also has borrowed money at low rates over the past few years.

Apple and the Irish government said they would appeal the ruling.

“We never asked for, nor did we receive, any special deals,” Apple chief executive Tim Cook said in a letter to customers. “We now find ourselves in the unusual position of being ordered to retroactively pay additional taxes to a government that says we don’t owe them any more than we’ve already paid.”

The tax rates described by the European Commission, as low as 0.0005 percent, are not accurate, Apple officials said.

“It is a completely made-up number,” said Luca Maestri, Apple’s chief financial officer.

The company paid $400 million in taxes to Ireland in 2014, Maestri said. “We were one of the largest taxpayers in the country, most likely the largest,” he said.

Apple, which has had operations in Ireland for decades, said the ruling could ripple throughout Europe and scare away companies from investing in that part of the world.

“The European Commission has launched an effort to rewrite Apple’s history in Europe, ignore Ireland’s tax laws and upend the international tax system in the process,” the company, based in Cupertino, Calif., said in a statement.

In Dublin, Ireland’s finance minister, Michael Noonan, denied that the country sidestepped E.U. tax rules, raising yet another potential flash point between Brussels and member states over the reach of regulations and oversight. Such questions helped tip the scales in June when Britain voted to leave the 28-nation bloc, and they have complicated transatlantic trade talks.

Appealing the decision “is necessary to defend the integrity of our tax system, to provide tax certainty to business and to challenge the encroachment of E.U. state aid rules into the sovereign member state competence of taxation,” Noonan said in a statement.

Across Europe, just how much — or how little — U.S. multinational firms pay in taxes is coming under increasing scrutiny.

French authorities recently raided the Paris headquarters of two U.S. corporate giants, Google and McDonald’s. And European authorities have accused the Netherlands of allowing Starbucks to avoid more than $30 million in taxes.

The Obama administration has repeatedly objected to these investigations, and the Treasury Department took the unusual step last week of issuing a 25-page report critiquing the European Commission’s investigations into alleged tax-avoidance schemes by U.S. firms. The investigations “undermine” agreements on international tax law and could hurt U.S. taxpayers, the Treasury Department said.

But it may not be able to stop them.

The European Commission’s actions “could threaten to undermine foreign investment, the business climate in Europe, and the important spirit of economic partnership between the U.S. and the EU,” a Treasury spokesman said in a statement.

The international scuffle centers on the more than $2 trillion in overseas profits that U.S. corporations have refused to bring back to the United States, where they would face a hefty tax bill. U.S. lawmakers and regulators have lamented the practice but have had little success in pressing the corporations to bring the money home. Apple has the largest stash of foreign profits — more than $200 billion — of any U.S. multinational company.

Now European tax authorities also are eyeing this money, international tax experts say. The profits often have been routed through low-tax European countries, potentially cheating other nations in which the companies operate, they argue.

Apple runs its European operations through Ireland, where the corporate tax rate is 12.5 percent, compared with the 35 percent statutory rate in the United States. But the European Commission found that Apple set up a system that allowed it to pay even less.

“Ireland granted illegal tax benefits to Apple, which enabled it to pay substantially less tax than other businesses over many years,” European Commissioner Margrethe Vestager, who is in charge of competition policy, said in a statement from Brussels.

Apple attributed its European profits to phantom “head offices” in Ireland, the investigation found. These offices “existed only on paper and could not have generated such profits,” the commission said. In fact, Apple recorded all of its European sales in Ireland rather than in the countries where the products were sold, the investigation found.

The “tax treatment in Ireland enabled Apple to avoid taxation on almost all profits generated by sales of Apple products in the entire” European Union, the commission said.

Brian Murphy contributed to this report.