The way the process works is, they study the case and decide which way they will "vote" in the decision. In this case, they came out 6-3. At that point the Chief Justice assigns one of the "6" to draft the opinion. When the draft is complete it is circulated among all the justices, then feedback comes in, it's re-written a few times, but the final tally generally does not change.
Anyone seeing this development and attributing it to "radical right-wingers" on the court is an idiot. RvW was a horrible decision - possibly the worst in the past 50 years. It is based on nothing, and takes a horrific leap into policy-making, which the Court has no right or power to do. It has NOTHING TO DO with abortion, or public policy on abortion. That aspect is left to the state legislatures, where it belongs.
The reason why Leftists in Congress have been so fixated on RvW when interviewing Republican nominees is because they have KNOWN ALL ALONG what a horrible decision it was, and how easily it could be overturned, given the right case.
Unfortunately for the public dialog, Democrats are too stupid to separate the Constitutional issue from the public policy issue. And of course there has never been sufficient support for the Leftists position on abortion to do anything legislatively. Which is why they constantly lie about polls showing that the public supports them. Liars.