(Picture from the Tiger Woods Foundation.)
Antitrust law is about more than market share. It's about using that market share to destroy competition. A monopolist must take an overt action against another company before the Justice Department is going to take a look at it.
IBM's mainframe monopoly in the 1950s was followed by IBM using its control of designs to keep competitors out of its market. AT&T's telephone monopoly in the 1960s was followed by its trying to prevent the creation of MCI. Microsoft's monopoly in the 1990s was followed by its bundling Internet Explorer to destroy Netscape.
You have to control the market and use that control to abuse other entrants before lawyers toss the monopoly grenade at you.
Let's put it this way. The Bill and Melinda Gates Foundation is large and powerful. It is bigger, relative to its foundation brethren, than any other charitable foundation ever.
Now Tiger Woods also has a foundation. It had an event near my home yesterday. It involved teens, and goal-setting. The only way for Tiger to have a beef against the Gates folks would be if they tied-up all the teens in the country, actually preventing Tiger from helping any.
I doubt Tiger would be upset in that case. It's only if he's prevented from helping anyone by the dominant foundation's actions that he might have call to become angry. It's not going to happen.
An open source monopoly is simply not a barrier to competition. Even if a GPL program had market dominance you could still compete with it.
The whole question is silly except for our friend Matt Asay's realization that foundations are a stable form of open source governance. I have no monopoly on that. I don't even have a business model for it. All I have is what Tiger would offer if the Gates Foundation solved all teenagers' problems.