Since they were joined in a civil union, the law may look on it the same way as a child born to a married woman—the father is assumed to be the husband unless other legal challenges are made.
She may have a chance if she moves where a state where such civil unions aren’t recognized, Virginia seems willing to recognize Vermont law. Other states may not be so cooperative.
That's the thing - Virginia not only does not recognize such unions, they are specifically excluded from being recognized or considered in any way during custody disputes. No wonder the woman thinks she can't get justice - the Virginia Supreme Court is breaking the laws of its own state. Running, if she has done so, is a tactical error. This should have gone to the USSC first.
Virginia does not recognize same sex civil unions or same sex marriages by its own laws or by state constitutional amendment, and yet, so far they are rolling over and not standing up for their own constitutional laws.
Even if "paternity" isn't assumed in this case, it's likely Jenkins adopted the child, which would make her legal position the same as any other stepparent adoption.
She may have a chance if she moves where a state where such civil unions aren’t recognized, Virginia seems willing to recognize Vermont law. Other states may not be so cooperative.
The civil union is already dissolved. I don't think Miller is likely to find a state that's willing to abrogate a child custody order from another state in favor of the biological mother; that would have implications for every custody case involving an adoptive parent.