Chief Justice Roberts seemed quite paternalistic based on his remarks during oral arguments in Schwarzenegger v. EMA.

MR. SMITH: Except that cigarettes are not speech, Your Honor. This is fully protected speech.
CHIEF JUSTICE ROBERTS: I know that cigarettes are not speech, Mr. Smith. Cigarettes are something that we have determined are harmful to children. The question is, you say the record doesn’t support the idea that these video games are harmful to
children. Some of us may conclude that it does.

I am guessing that “some of us” includes the Chief”

Chief Justice Roberts next draws an analogy to 8th amendment jurisprudence, and alcohol laws, where minors are treated differently from adults.

CHIEF JUSTICE ROBERTS: We draw that kind of line of course in the death penalty area, don’t we? Between 18-year-olds? You are under 18; you can’t be sentenced to life without parole; if you were over 18 you can.
MR. SMITH: You do draw that line, Your Honor.
CHIEF JUSTICE ROBERTS: And we do it for drinking; we do it for driving.

Interesting that Roberts cites Graham v. Florida, in which he concurred.

In Graham, Roberts was amenable to the notion that juveniles could be less culpable than adults:

The second line of precedent relevant to assessing Graham’s sentence consists of our cases acknowledging that juvenile offenders are generally —though not necessarily in every case—less morally culpable than adults who commit the same crimes. This insight animated our decision in Thompson v. Oklahoma , 487 U. S. 815 (1988) , in which we invalidated a capital sentence imposed on a juvenile who had committed his crime under the age of 16. More recently, in Roper, 543 U. S. 551 , we extended the prohibition on executions to those who committed their crimes before the age of 18.

…

Justice Thomas disagrees with even our limited reliance on Roper on the ground that the present case does not involve capital punishment. Post , at 26 (dissenting opinion). That distinction is important—indeed, it underlies our rejection of the categorical rule declared by the Court. But Roper ’s conclusion that juveniles are typically less culpable than adults has pertinence beyond capital cases, and rightly informs the case-specific inquiry I believe to be appropriate here.

Roberts seemed to be the only voice opposed to striking down this law. I do not think he will vote to strike down the law, but we can probably expect a concurrence penned by the Chief, or if you ask Dahlia Lithwick and Barry Friedman, a Houdini-esque attempt to shift a likely 9-0 opinion to the Right.