Earlier quoted context omitted.
1. It isnt clear ICE agents have any legal authority to demand a judge tell them anything. 2. It is highly likely this is an official act, since it would be taken on behalf of court, so the immigrant can give, eg. testimony in a case. A "private act" here would be the judge lying in order to prevent their deportation because they as a private person wanted to do so. It seems highly unlikely that this is the case.
I updated my post with new information from the updated article, and in the context of that I think you're pretty much right. It sounds like the judge basically said "you need permission to arrest someone in the middle of my hearing, go get it" and then didn't change anything about the process of their hearing while that permission was being obtained. This was definitely not them being helpful , but I'm incredibly do…
The allegation is that the judge got upset that ICE was waiting outside the courtroom, sent the law enforcement officers to the chief judge's office, and then adjourned the hearing without notifying the prosecutor and snuck the man with a warrant out through a non-public door not normally used by defendants.
If those facts are accurate, it sure sounds like obstruction. Judges have to obey the law just like everyone else.