The UK Prime Minister invited Donald Trump to visit the country; Queen Elizabeth did not intervene to cancel that invitation.
The Queen does not intervene in political disputes.
On 7 June 2017, the Patriot Beacon reported that Queen Elizabeth II had publicly contradicted London mayor Sadiq Khan by "refusing" to retract an offer of an official state visit to President Donald Trump after Khan had suggested calling it off:
After the terrorist attack in the United Kingdom over the weekend, London’s Mayor Sadiq Khan called for Donald Trump’s upcoming trip to his city to be cancelled based on what he said was the American president’s “alarmist” reaction to the attack. Days later, however, Queen Elizabeth shut Khan down in a big way.
Breitbart reported that the Queen refused to retract her invitation for Trump to meet with her at Buckingham Palace, ruining Khan’s plans to stop Trump from traveling to London.
“[The president] appreciates Her Majesty’s gracious invitation,” White House Press Secretary Sean Spicer said when asked if Trump would still meet with the Queen.
This sends a huge message to Khan, as it shows that the Queen has no intention of submitting to him and the agenda he is inflicting the city of London.
A spokesperson for Buckingham Palace confirmed to us that a formal invitation to visit the UK had not been extended to Donald Trump by Britain's royal family; Sean Spicer appears to be mistaken in referring to "Her Majesty's gracious invitation."
Queen Elizabeth has not refused to retract her invitation, because she never formally extended one. Nor has she made a public statement on the matter, either before or after Sadiq Khan called for Trump's official visit to be cancelled. The Queen never intervenes in political disputes in any case, and her silence in the two days following Khan's remarks is entirely in keeping with normal practice and should not be interpreted as meaningful in any way.
In January 2017, UK Prime Minister Theresa May did extend an invitation to the US President to visit, but it's unclear whether this will be a state visit or official visit. In the UK, a state visit is one where the head of state (Queen Elizabeth) formally hosts a foreign dignitary (in this case, Donald Trump). An official visit means the foreign dignitary is invited to visit the UK, but is not formally hosted by the Queen.
The episode has proven controversial, with one experienced former diplomat, Peter Ricketts, writing at the time that May's invitation had put the Queen in a "very difficult position." In an interview with Channel 4 News on 5 June 2017, London's mayor was asked if he thought Trump's state visit should be called off:
Well, my position remains the same - I don't think we should be rolling out the red carpet to the President of the USA in the circumstances where his policies go against everything we stand for.
— Channel 4 News (@Channel4News) June 6, 2017
Apart from the fact that the Queen has not "refused to retract" her invitation to Trump (because she has not extended such an invitation), there are a couple of other inaccuracies in the Patriot Beacon article. For example, it cites the conservative web site Breitbart as reporting "that the Queen refused to retract her invitation for Trump to meet with her at Buckingham Palace."
The 6 June 2017 article does not say that; instead, it simply quotes White House Press Secretary Sean Spicer's response to a question on Sadiq Khan's call for a cancellation:
Press Secretary Sean Spicer was asked at the daily White House press briefing on Tuesday about London Mayor Sadiq Khan's saying that President Donald Trump's visit to the UK should be cancelled after the president criticized the mayor's remarks about the terror threat there.
Khan said there was “no reason to be alarmed” by an increased police presence in London following yet another terror attack on Saturday on the London Bridge.
“[The president] appreciates Her Majesty’s gracious invitation,” Spicer responded, referring to the invitation extended to Trump by Queen Elizabeth.
The Breitbart article also does not describe Queen Elizabeth as refusing to retract her invitation or taking any action (or conspicuous inaction) following Sadiq Khan's remarks the day before.
The Patriot Beacon article claims that Khan had called for Trump's visit to be cancelled on the basis of Trump's "alarmist" response to the London Bridge attack. Khan never described Trump's response as alarmist, but said instead that his call for the visit to be cancelled was because Trump's policies "go against everything we stand for".
Khan had already opposed an official visit by Trump to the UK in response to the US President's proposed immigration ban, and long before the June 2017 terrorist attack. In an op-ed for the London Evening Standard on 30 January 2017, the mayor wrote:
The Prime Minister must be clear with President Trump that his actions are unacceptable for a liberal, open democracy. And we should not be seen to be endorsing them.
That is why we must now rescind the offer of a full state visit for President Trump — until this ban is lifted. I don’t believe the people of London will support rolling out the red carpet until this happens.