And the 36-year-old royal insisted the tournament is about far more than just winning.
He added: "It's not about the medals. It's not even so much about the winning, as far as most of us are concerned you guys have already won.
"These games are so important, whether it's the Warrior Games, whether it's the Invictus Games, it is ultimately the reminder of service, both at home and overseas.
"It is about caring for our physical as well as our mental fitness.
"And that, again, is not just those who wear the uniform, but every single one of you.
"Because as we know, once served, always serving and that goes across the whole family network."
And the First Lady, 70, praised Harry as she reflected on the "humanity" that shone through after the September 11 terror attacks of 20 years ago.
She said: "Twenty years ago our world broke apart and the shockwaves of 9/11 continue today, but as the President and I visited the September 11 memorials this weekend I was reminded of the humanity that shined through the inhumanity of that day.
"There was a call to defend the values we hold dear as we waged a global war on terror and a courageous group of men and women stepped forward to say 'Me, I will go'.
"And that includes you, Harry. You live by a simple principle, serve together, recover together."