Not quite. Microsoft had market power in the desktop OS market where they leveraged their dominance of that market to force their partners to give them preference in the browser market.
Apple doesn't participate in a "mobile OS market". It's simply a feature of their mobile devices.
Contrast that with Google, who enters into agreements with their horizontal competitors to limit competition in various markets by leveraging the Android brand to give preference to Google services.
Apple has similar power in mobile OS. AT&T, Best Buy, etc. are restricted from loading their own or someone else's app stores, browser engines, etc. on iPhones they sell and as such, Apple is "forcing" their retail partners to give Apple preference in the app store, browser engine, etc. markets.
Apple does participate in the mobile OS market. They just happen to sell (license) it to consumers or to their retail partners as a package with the iPhone device. Apple also provides new iOS software separately for a period of time to iPhone owners.
That's simply not true from a market perspective. Just because two pieces of software are based on the same open source project does not mean that they aren't competing with each other.
Apple's iOS and Google's Android are the two major players in the mobile OS market. In the U.S., iOS happens to have around 55% to 60% of that market.