I'm not a lawyer but the basics of U.S. antitrust law are pretty easy to sort out for those who wish to. And most people can read case law to gain a more nuanced understanding of how it applies if they're really interested.
The answer to your first question is: No, there isn't any provision of U.S. antitrust law which specifically prohibits not allowing side loading. As is the case in many areas of law, the actual statutory provisions are quite general and don't refer to specific practices - e.g., not allowing side loading. Case law builds up around those general provisions and provides greater clarity on what they mean and how they might be applied to different situations, and then based on such guidance from case law case-by-case decisions are made as to whether particular practices by particular parties are illegal.
There are two kinds of Sherman Act violations at issue here. (There are other provisions of U.S. antitrust law, but these Sherman Act violations are the ones at issue here.) There's a Section 1 violation and a Section 2 violation, and both sections are generally worded:
Section 1:
Section 2:
So, looking at Section 2, it makes it illegal to monopolize or attempt to monopolize any part of trade or commerce. The wording is very general and doesn't refer to specific practices which might constitute monopolization. But case law tells us more about what constitutes illegal monopolization. In U.S. v Grinnell Corp (1966), e.g., the Supreme Court said:
Put another way, a violation of Section 2 requires (1) monopoly power, (2) exclusionary conduct and (3) a casual connection between the exclusionary conduct and the monopoly power. Monopoly power, by the way, doesn't necessarily mean complete control of a market. Something considerably less, even 60% market share, can constitute sufficient market power for Section 2 purposes.
So to answer the query in your second paragraph, Apple not allowing side loading of apps in iOS could (thought I'm not suggesting it should) constitute the exclusionary conduct required for a Section 2 violation. The argument would be that Apple unlawfully maintains its monopoly power in the iOS app distribution market (I won't get lost in antitrust market definition here, that's another aspect of the consideration) by creating technical and contractual barriers to competition. Its iOS software is designed to prevent, not always but generally, the loading of apps other than through the App Store. And Apple requires would-be iOS developers to agree not to distribute their apps other than through the App Store.
I want to be clear that I don't think Apple should be found to have violated antitrust laws in this way. But that is, more or less, the Section 2 argument you are asking about. It's why Apple's efforts to block side loading might, without regard to the 30% commissions it charges, be an antitrust problem. If it's decided that iOS app distribution is in itself a relevant antitrust market, then Apple could easily be found to have engaged in illegal exclusionary conduct meant to help it maintain its monopoly in that market. One question would be, does the anticompetitive conduct (i.e the technical and contractual barriers to competition) have procompetitive justifications? In other words, is Apple doing those things just to block competition in iOS app distribution? Or is it doing those things for other, legal, reasons - e.g., to provide better security for its customers.
