Did the Commissioner/DOJ present the Facebook picture as evidence, or did the judge find it on his/her own? I would think that the judge taking notice of this would be unethical ex parte "communications."

Yes, but only speech that threatens harm or intimidates can be constitutionally regulated. See Virginia v. Black, 538 U.S. 343 (2003) (holding that while prohibiting cross-burning with the intent to intimidate is constitutional, making cross-burning prima facie evidence of intent to intimidate then criminalizing cross-burning with the intent to intimidate is not).

Saying "the Holocaust didn't happen" is clearly a factual misstatement, but that doesn't make it intimidating or threatening. There's a line between regulating hate speech and regulating speech that incites to dangerous action.

However, I believe there's a huge difference between a law that criminalizes Holocaust denial and a policy of Facebook, a private entity, to not permit Holocaust denial using its resources. I think that a Facebook ban would be entirely appropriate.