Shutting down Twitter to stop them implies the Brazilian judiciary doesn't have the ability to sanction them under Brazilian law.
On a side note, the contents an international corporation publishes can violate the law of a country even when they are not posted by citizens of that country. When they do so, and don't take down the content, they will be shut down. It works like that in every country, including the US. It pains me having to point that out since it's so obvious.
For instance before the internet a Saudi dissident based in the UK used to send faxes to phones in Saudi Arabia that were critical of the Govt. I don't think the Saudi govt blocked access to phone calls from the UK on account of that.
If Brazilian enterprises are not allowed by law to trade with Twitter that does not amount to shutting it down. All you've done is block the usage of a service by most law abiding citizens which many of them probably rely on for all or a part of their living.
Twitter is operating under US law and Brazilian law only applies to their Brazilian subsidiary. If their Brazilian subsidiary has been has closed down because their parent company is not complying with Brazilian law then Brazilians are free to deal directly with the parent company under US law and jurisdiction.
Which is a pretty stretch from an older law, that was made to companies operating "phisically", opening offices, having workers and etc.
Twitter closed their offices in Brazil, so they don't need to do it anymore.
If we take the interpretation this judge is using with Twitter to everything, 99% of the internet will need to be blocked in Brazil.
Actually, millions of people are, and senators are seeking this judge's impeachment.
That's the point, they don't. There's nothing in the law for him to back his decisions. But we just hope for our senate to take action.
In the end of the day, people just care if their political agenda is being met, it doesn't matter if it throws the country into a dictatorship.
For instance in the UK the government can stop local media from publishing news that is considered libellous or defamatory, or may unjustly influence court cases, but they can't stop citizens from accessing that information if it is published in a news outlet outside the country.
No one is questioning why the Brazilian judiciary should have that power. It seems rather excessive and an infringement of Brazilians rights, though that is not surprising when we live in a era when governments have taking upon themselves to decide what their citizens rights are, when human rights are supposed to be intrinsic, ie those that all fauna on the planet do.