There is much at stake for both sides.

Ireland’s low tax rate and other financial incentives have attracted hundreds of foreign companies, providing thousands of jobs and hundreds of millions of dollars of annual investment. Any restrictions on how Ireland can use its tax code, experts say, could seriously hamper the country’s growth and leave Apple with a multibillion-dollar tax bill to pay. The company plans a separate appeal.

For European officials, the Apple tax ruling is the most visible of a series of cases against American companies accused of using Europe’s tax system to reduce their tax burdens unfairly. Lawmakers worldwide are trying to clamp down on complex tax structures that have allowed many multinational companies to significantly lower their tax liability.

“We need a change in corporate philosophies and the right legislation to address loopholes and ensure transparency,” Ms. Vestager said when announcing the tax clawback against Apple in August.

Already, Starbucks has been fined roughly $34 million for an preferential tax deal in the Netherlands. McDonald’s and Amazon are likely to face similar charges in Europe in the coming months. All the companies deny they have breached Europe’s competition rules.