U.S. District Judge Yvonne Gonzalez Rogers rejected Apple Inc.’s attempt to delay her September injunction that would loosen some of the App Store rules for software developers.

During a hearing Tuesday, Apple sought to stay her order that would force the company to allow developers to communicate with users inside their apps about alternative payment methods outside of the App Store.

She...

U.S. District Judge Yvonne Gonzalez Rogers rejected Apple Inc.’s attempt to delay her September injunction that would loosen some of the App Store rules for software developers.

During a hearing Tuesday, Apple sought to stay her order that would force the company to allow developers to communicate with users inside their apps about alternative payment methods outside of the App Store.

She originally gave Apple until Dec. 9  to change its App Store and said Tuesday that a stay on the injunction would delay its implementation for years.

“Apple believes no additional business changes should be required to take effect until all appeals in this case are resolved. We intend to ask the Ninth Circuit for a stay based on these circumstances,” an Apple spokeswoman said in a statement.

The order threatens to make it easier for developers to steer users outside the App Store when they sign up and pay for services and beyond Apple’s in-app purchase system that captures a share of digital revenue generated inside of apps.

Judge Gonzalez Rogers in September ordered the change as part of a broader antitrust lawsuit brought by “Fortnite” maker Epic Games Inc. that dealt with the power Apple holds over software developers. The videogame maker wants access to Apple’s mobile devices but has fought the company’s requirement that they use the tech giant’s in-app payment system, which in turn collects fees of as much as 30% of digital revenue.

The case put a spotlight on Apple’s App Store business, which it doesn’t break out in its financial results. Records revealed during the trial suggest that in recent years one out of five dollars of the company’s overall operating profit came from the App Store. Most of those sales came from videogames, such as “Fortnite.”

Apple has said the operating margins discussed during the trial were flawed and as a result are too high.

Apple mostly won the case, which included a 16-day bench trial before Judge Gonzalez Rogers, who rejected Epic’s request to force Apple to allow software to be downloaded outside of its App Store. Apple had argued the App Store and its app review process protects users from cyber threats.

Epic has said it will appeal the verdict. On the anti-steering issue, Apple has said it is appealing the matter to a higher court, too, and, in turn, was seeking a stay on the injunction.

“It’s exceedingly complicated,” Mark Perry, a lawyer representing Apple from Gibson, Dunn & Crutcher LLP, told the judge. “There have to be guardrails and guidelines to protect children, to protect developers, to protect consumers, to protect Apple. And they have to be written into guidelines that can be explained and enforced and applied.”

He noted that Apple is already working on a narrow change to the App Store that would allow media apps, such as Netflix Inc.,

to create in-app links to sign-up pages on those companies’ websites. The move is part of an agreement with the Japanese antitrust regulators to end an investigation announced in September before the judge’s ruling was issued.

The company has also announced a proposed settlement in a separate class-action lawsuit in which Apple agreed to drop restrictions preventing  developers from using certain information captured in their apps to communicate with users outside of the app.

Epic’s lawyer argued against the stay, pushing back against Apple’s request for more time to solve the matter. “Your Honor has found that Apple does nothing unless it is forced to do it,” said Gary Bornstein, Epic’s lawyer from Cravath, Swaine and Moore.

On Tuesday, the judge didn’t seem moved by Apple’s argument, saying at the beginning of the hearing that she wasn’t inclined to grant the stay.

“You haven’t asked for additional time,” she later told Apple’s lawyer via a video link. “You did not ask for a few months, you didn’t ask for six months, you didn’t ask for a limited amount of time. You asked for an across the board stay, which could take three, four or five years.”

Later in her order denying the request, she wrote that Apple hadn’t provided any credible reason for her to believe the injunction would cause the “professed devastation.” Furthermore, she wrote, users are used to linking from an app to web browser.

“The Court can envision numerous avenues for Apple to comply with the injunction and yet take steps to protect users, to the extent that Apple genuinely believes that external links would create issues,” she wrote. “The Court is not convinced, but nor is it here to micromanage.”

She added that her original order doesn’t prohibit Apple from banning third-party in-app purchase systems. “The Court did not enjoin that provision but rather enjoined the prohibition to communicate external alternatives and to allow links to those external sites,” she wrote.

Write to Tim Higgins at Tim.Higgins@WSJ.com