The Ninth Circuit's decision depended on a right they said the father had to determine his daughter's religious upbringing. I would think that that right goes with legal custody, not physical.
At least in Wisconsin, sole legal custody means that ONLY the custodial parent makes the major decisions in the child's life--choice of school, choice of religion, where the child lives (within the restrictions of the law) etc. And that would probably extend to being a litigant in a lawsuit, or being used by a parent to be a litigant in a lawsuit. From the parts of the interview I saw with the mom last night, the mother wasn't even told about the lawsuit. I don't know at which point she learned of it--I may have been when the Court of Appeals issued its decision. Given the backlash to the lawsuit, Newdow may well be endangering his child.
I hope that some of California's legal beagles chime in on this one.