> Courts do not require a literal monopoly before applying rules for single firm conduct; that term is used as shorthand for a firm with significant and durable market power — that is, the long term ability to raise price or exclude competitors. That is how that term is used here: a "monopolist" is a firm with significant and durable market power. Courts look at the firm's market share, but typically do not find monopoly power if the firm (or a group of firms acting in concert) has less than 50 percent of the sales of a particular product or service within a certain geographic area. Some courts have required much higher percentages. In addition, that leading position must be sustainable over time: if competitive forces or the entry of new firms could discipline the conduct of the leading firm, courts are unlikely to find that the firm has lasting market power.
The US doesn't have antitrust authority for the world, only for the US. iPhone has had 60% market share (or similar) for a long time now, so it's fair to consider that Apple has significant and durable market power in mobile phones.
Is it a complete monopoly? No, but it doesn't need to be.
From a very brief skim of the claims, the clearest one that stands out to me is the one about smartwatches. If Apple does provide better integrations to Apple Watches than 3rd party watches, that's pretty clearly 'tying' which is prohibited when using a market dominant product to create market dominance in a new market (smartwatches). OTOH, it wouldn't have been a big deal if the Microsoft Band had better integrations than other watches on Windows Phone, because tying is allowed without market dominance.
[1] https://www.ftc.gov/advice-guidance/competition-guidance/gui...