Not in the US they don't; it's a minority share, split across many different manufacturers who all have their own flavor of Android, their own (crappy) app stores and pre-loaded crapware apps, etc. Apple has a clear majority of the US smartphone market (and it's a vertical monopoly, with Apple controlling the phone hardware and the app store and not allowing any alternatives), but no one's looking at breaking them up.
>I think these two being open source is a major reason why they have been so successfully adopted.
iOS isn't open-source, and it has a commanding majority of the US smartphone market.