Yes, the US government should be tasked with stopping fake news by US-based news outlets. The only way for a functional democracy to exist is to have a well-informed electorate that knows the facts about issues that are occurring. The government should make sure that fake news is called out.
No, because it's not the primary duty of the government to prevent or say what is "fake", or "real" news. It's the responsibility of each one of us to do our research by looking to multiple sources, checking the credibility of those sources, before forming our opinions. Folks, please don't just rely on social media and/or one source of information. Didn't they teach us in school when writing a paper, that having multiple resources is helpful when informing others and ourselves?. How else are we supposed to be a democracy?
Nowhere does the U.S. constitution, or the slew of amendments and laws that govern the country call for the government to control news. In fact, the opposite is true--they are forbidden from stopping a free press. While outrageous and undoubtedly unethical, fake news stories should be handled by news agencies themselves. The AP should step in and demand accuracy in reporting, including debunking fake news stories as they arise.
I think anyone who agrees with that question needs to read the first amendment. That is how Trump is getting away with his public lies. News outlets need to be help accountable by the public. It is tough with the Internet, but eventually the truth does come up about these things.
The United States constitution protects freedom of speech and freedom of the press. The United States government does not get to decide which news is real and which news is fake. It is up to citizens to make up their own minds as to what to believe in the news.