Of course, the federal government's frontline argument is that the minimum essential coverage requirement is a valid exercise of the commerce power. And its argument on this score has two distinct components, either of which would independently justify the individual mandate: (1) the decision as to whether to purchase health coverage is an economic activity, and thus falls within the commerce power under the third prong of United States v. Lopez (activities substantially affecting interstate commerce); and (2) even if this "economic decision" is not considered an "economic activity," Congress is not limited to regulating "economic activities" when reaching that conduct (whether it is an activity or passive inactivity) is integral to the functioning of a broader regulatory scheme, which scheme plainly regulates an interstate market. (This latter argument typically relies on Wickard, Raich, and the Necessary and Proper Clause.) That is, the minimum coverage requirement is justified as necessary to the broader regulation of the health insurance and health care services markets, especially those imposing the guarantee issue and community rating requirements on the insurance industry.
In the Liberty University brief, DOJ seems to have slightly altered its framing of activity being regulated, which potentially affects both of these arguments. Here is the first subheading in the brief:
The minimum coverage provision regulates the practice of obtaining health care services without insurance, a practice that shifts significant health care costs to other participants in the health care market.
In other words, the regulated activity is not the "economic decision" of whether to acquire coverage or how to finance care (as three district courts have now held in upholding ACA 1501(b)). Rather, the regulated activity is indeed an activity, and by anyone's lights: the activity of acquiring health care services without insurance.
The challengers will of course respond that many of the uninsured will not actually obtain medical services, so the regulation reaches beyond those persons that Congress has the jurisdiction to touch. But then the analogy to Wickard and Raich is even stronger: Congress has no obligation to exempt out those individuals who are in a class that, as a class, are plainly engaged in interstate commerce. And the uninsured, as a class, consume roughly $50 billion in uncompensated care annually.
The better retort, I think, will be that the minimum coverage provision does not actually regulate the activity obtaining of health care services. Rather, it regulates a different act (or non-act): obtaining coverage. The advantage in this shift for DOJ, though, is that it potentially takes the issue out of the realm of the activity-inactivity distinction, and instead frames it as a matter of whether it is appropriate to mandate coverage as a regulation of the activity of obtaining health care services without insurance. The Supreme Court has always been deferential to Congress's judgment as to the appropriateness of means.
Again, this may be much ado about some small semantics. But then again, what is constitutional doctrine if not a complex game of words and phrases, with only a tenuous connection to empirical reality?