A lawsuit making the provocative argument that one of the biggest social-media platforms should be subject to the First Amendment got a hard look from a federal appeals court in Seattle on Tuesday.

During arguments before the Ninth U.S. Circuit Court of Appeals, Prager University, a nonprofit that produces bite-size explainer videos promoting conservative ideas, argued Google’s YouTube violated its First Amendment rights and misled its users when it limited access and demonetized dozens of its videos.

The...