False.
America was found by an Italian of the name Americo Vespucci. He did not take part in any British expeditions. In matter of fact British never discovered Americas.
And false again, not all of the US territory was part of the British empire, in matter of fact, not even one sixth of it.

What he means is, most Americans have a British background either being related to a British person or being related to someone Britain had enslaved.
Basicly all of America (USA and Canada) was controlled by the British Empire. :poke: