One might argue that forcing you to use their app store constitutes illegal tying. The iPhone could easily be made to run apps from competing stores (as is possible with Android). Don't think about competitors that do exist on other platforms (like Play Store), but instead think of competitors that were never allowed to exist on iOS.
It's not an implausible argument! It's already been successfully argued that "jailbreaking" is legal. I just don't know how easy the argument would be to make, and what the counterarguments would be; it's hard to think of any very good precedent for this. (Nintendo's old "lockout chips" are the closest I can think of, maybe?)
Jailbreaking is legal because you purchased the hardware and can use it as you please.
This is very different from compelling companies to develop and support channels for installing third party software.
Well, that's the other side of the argument, right? Apple has developed most or all of the technical underpinnings for this already, as demonstrated by the ability to do enterprise provisioning and install apps through TestFlight. It wouldn't require an amazing engineering feat for them to bring a version of Gatekeeper to iOS that let you install signed applications from anywhere, not just their app store -- the plumbing is basically already in place. The question is really whether there's a legal case for that.
While I've generally come down more on your side of that argument in the past, I'm not longer quite so sure; Apple's control over the iOS app ecosystem as things are now has no precedent in the history of computing I'm aware of, and I could certainly see a plausible case made that this control does ultimately harm consumers by making it effectively impossible to install non-Apple-sanctioned software on iOS devices. Both "if you didn't want to accept that you shouldn't have bought an iOS device" and "but you can technically jailbreak the device if you find a way Apple hasn't cut off yet" are plausible defenses, too, but neither one seems to me like a legal slam-dunk.