No, not remotely.
It may be true that a particular company has a stranglehold on browser choice on your particular phone, but that is completely different from having an anticompetitive strangehold on the market as a whole, which is what matters from an antitrust standpoint. I'm not the biggest fan of capitalism but the market seems to be functioning rather well. Not perfectly but well.
Just a refresher: IE had 96% market share at its peak, and Microsoft were guilty of quite a few other anticompetitive practices as well w.r.t. Windows itself, more or less literally penalizing OEMs from shipping any other desktop OS.
https://en.wikipedia.org/wiki/Browser_wars
Competition is thriving in the mobile space. Safari, Chrome, and derivatives of Chrome are competing with each other and they are well-performing, aggressively maintained and updated pieces of software.
Sure, there are plenty of technical reasons to be annoyed with either one of them. Mobile Safari sure is slow to adopt certain standards, etc. And don't get me wrong -- it is extremely regrettable whenever browser choice is restricted. It sucks, I hate it.