According to Debrett's, official protocol / etiquette dictates that in carrying out a toast to the Queen, the toaster will rise and request the audience's attention. Once accomplished and the guests are standing, the host raises his or her glass and recites the title of the person being toasted, without any other words or music. So for example:
"Ladies and Gentlemen, Her Majesty The Queen." or
"Ladies and Gentlemen, Her Majesty The Queen, Head of the Commonwealth."
"Ladies and Gentlemen, Her Majesty The Queen, Duke of Lancaster" (in Lancaster)
etc.
The same official etiquette exists in the US. The US armed forces toast to the president is simply "The President of the United States." So this is not something the President should be unfamiliar with.
The error was that the President should not have planned to say anything else after reciting the toast. Even the orchestra knew that, and that's why they struck up the anthem the moment they heard those words.