The legal battle between Apple Inc. and the creator of “Fortnite” is doing more than just fraying a nearly decadelong relationship. It is also prompting other tech heavyweights such as Microsoft Corp. to step in, as the feud has escalated to a point where it could upend business at makers of videogames, Hollywood productions and virtual-reality simulations.

Apple, the world’s most valuable company, earlier this month threatened to pull Epic Games Inc.’s developer credentials—which are required to release software on Apple…