BRUSSELS—“Fortnite” game developer Epic Games Inc. has filed an antitrust complaint against

Apple Inc.

AAPL -1.61%

in the European Union, broadening its continuing legal battle after filing similar lawsuits in the U.S., Australia and the U.K.

At the core of the legal dispute is how much control and revenue share technology giants should have in relation to popular apps. The online game “Fortnite” was kicked out of both Apple’s App Store and

Alphabet Inc.’s

Google Play Store last year after Epic introduced a payment system that effectively cut off both companies from the 30% share of users’ spending that they had charged Epic.

Epic has sued both Apple and Google for alleged monopolistic behavior in the U.S., and Apple countersued.

In Europe, Epic’s legal complaint joins plaintiffs including music-streaming service

Spotify Technology SA

that have prompted a formal investigation into Apple’s alleged anticompetitive behavior.

“What’s at stake here is the very future of mobile platforms,” said Epic founder and Chief Executive Officer

Tim Sweeney.

“We will not stand idly by and allow Apple to use its platform dominance to control what should be a level digital playing field.”

Mr. Sweeney said that while the legal case filed in Europe targets Apple, “the broad outlines…are equally applicable to Google, though the timing may be different.”

Originally published Oct. 1, 2020: Apple’s stock-market value hit a new record this year, but its longstanding disputes with app developers are bubbling over into public view. WSJ explains why high-profile companies like Epic Games at odds with App Store rules. Video/illustration: Jaden Urbi/WSJ

Apple couldn’t be immediately reached for comment. The company has previously dismissed Epic’s lawsuits as “a basic disagreement over money” and said the complaints that prompted the EU’s antitrust investigation last year were baseless.

The European Commission—the EU’s executive arm—didn’t immediately comment on Epic’s complaint.

The similar cases Epic has filed against Apple in the U.K. and Australia also allege abuse of dominance. The game maker’s U.K. lawsuit last month against Google likewise accuses the search giant of abuse of dominance and anticompetitive behavior on its Play store.

Google said in response to the lawsuits in the U.S. and U.K. that Epic violated rules applying to all developers that use the Play store, adding that the company would continue discussions with Epic about bringing “Fortnite” back to the store.

In Europe, Epic claims that Apple blocked “Fortnite” updates in retaliation for Epic having given users the option of paying directly via the app instead of with Apple’s own payment system, Apple Pay. When Apple launched its own games-distribution service, Apple Arcade, it barred competitors from doing the same, Epic alleged in its complaint.

Apple’s operating system, iOS, “should be open to competing stores,” said Mr. Sweeney, adding that iPhone users and app developers shouldn’t be restricted to using Apple Pay but that other payment systems should be equally permitted. When launching the investigation last year, the European Commission said it would look into how Apple Pay is the only service allowed on the no-contact payment system built into Apple devices.

Write to Valentina Pop at valentina.pop@wsj.com

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