No, the judge cannot. Again, there is no standard. There is no grounds by which he can say it’s perjury.
I can imagine some situations, for instance if he went on Facebook and clicked "male", or wrote a blog that said "I am a man...". If he had separate self-ID as a man, that could be used as evidence that the self-ID as a woman was a lie.
But that's neither generally the case (not everyone makes public declarations of their gender)*, and the other day-to-day activities that he engages in don't implicitly make that declaration either. Even using the male locker room on occasion could be explained as that he's worried about transphobic reactions if he uses the woman's locker room.
So, aside from the above minor caveat, I'm also waiting for Archie's explanation of what evidence could be brought.
*ETA sometimes I jump ahead, I meant to also say that someone who had made that gender change might be motivated to explicitly avoid such public declarations. It wouldn't be that hard.
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