In the speech he says "then our Empire beyond the seas, armed and guarded by the British Fleet, would carry on the struggle, until, in God's good time, the New World, with all its power and might, steps forth to the rescue and the liberation of the old." By the New World did he mean the United States?
That segment of his speech is particularly interesting for a number of reasons. He was indeed referring to the United States, but he's including the rest of the Empire too.
The American government wanted him to include this part. The USA of course was still neutral at this stage – Churchill’s initial draft included a reference, which he later deleted by hand, to America's ‘strange detachment’ in the face of the Nazi menace. However, President Roosevelt made clear via secret channels that he wanted a commitment from Britain that even if she were defeated she would not surrender her fleet but would send it to South Africa, Australia, Canada and other parts of the Empire.
If this were done, Roosevelt promised, American intervention could be expected to follow quickly. Churchill including this statement, therefore, was a nod that America was on the allied side.
Further reading: The Roar of the Lion: The Untold Story of Churchill's World War II Speeches (OUP Oxford, 2013)