Bob is watching Youtube videos and sees a video of a judge trying to enforce their authority in a public building. This enrages Bob and he begins to call the judge a slew of profanities online. The judge happens to see Bobs comments online. Can the judge hold Bob in contempt or bring any other legal action against him under his authority as a judge even thou its a different state?
Contempt powers are rather broad, but not that broad. A judge has power to find anyone in the courtroom in contempt, if that person acts in violation of he rules or of the judge's instructions.
A judge has the power to find any party to the case in contempt if that person violates orders from the judge, unless there is some overriding circumstance, for example if the person's action was an exercise of a constitutional right, or if the judge's order is held (by a higher court) to be unreasonable.
A judge will have the power to issue orders to people not otherwise parties if this is needed to preserve the judicial process, such as "gag orders" to prevent bias from contaminating a trial. These often have conflicts with first amendment rights or other rights, and there have been various cases on exactly how far a judge can go in such matters. But if the Judge is within those rules, violations of the judge's orders may be punishable as contempt.
But an ordinary individual who is simply expressing disapproval of the judge, even in a highly disrespectful fashion, neither in nor near the courtroom, should not be subject to a contempt citation, and few judges would try to invoke such a power.
Note that a judge can always issue a contempt decree or citation, and may be able to have the person jailed, if the person is within the court's jurisdiction. But if the citation was an abuse of discretion, it can be overturned and the judge may be subject to a damage suit, in an egregious case.
Charles Rembar in The Law of the Land wrote of one case where the judge had had a bailiff purchase coffee during a recess, and found it not at all to his liking. He had the coffee-shop owner arrested, and brought into his presence, and proceeded to scream at him over the poor coffee. The owner later sued the judge, and received a 6-figure damage award. as Rembar put it "The price of coffee is going up all the time, and it seems that the jury felt that Coffee with scream was extra."