Unbroken, which was made available to stream and download after midnight, was recorded at the world-famous Abbey Road Studios in London in February.
It will "touch many hearts and be a light during this troubling time", one singer in the choir said.
Bon Jovi, whose parents both served in the US Marine Corps, originally wrote and released the track last year as a tribute to veterans living with post-traumatic stress disorder.
Alongside Harry and 12 members of the choir, the US rocker re-recorded Unbroken across two days in aid of the Invictus Games Foundation.
The foundation oversees the development of the Invictus Games, which was postponed this year due to the coronavirus pandemic.
Susan Warner, a member of the choir who was seriously injured during an Afghanistan deployment in 2009, said the song offers hope to veterans during the Covid-19 outbreak.
The 61-year-old said: "We hope the single will touch many hearts and be a light during this troubling time for so many people around the world. It just feels like it is so important.
"In life you never know what might open up or what is going to happen. This is a very good example. There we were in Abbey Road Studios and we were singing Unbroken. Then, a month later, here we are."
The visit also saw the duke, Bon Jovi, Warner and another member of the choir step on to the famous zebra crossing to recreate The Beatles' Abbey Road album cover.
Warner, who competed in the 2016 Invictus Games in Orlando, Florida, said Harry's presence was an "honour and privilege".
She added: "The hope he brings to us all at this time, to be on the zebra crossing and re-enacting history where The Beatles stood - it was just such an honour and privilege.
"To be of service, to be of duty, to be able to again help someone to reach out."
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Harry shares behind the scenes look as he joins Bon Jovi at Abbey Road
Bon Jovi said he had thought plans to record the single had been scuppered by Harry's decision to move to Canada with the duchess.
He said: "In light of everything that was happening here now I thought 'Oh well, there goes that great idea.'
"And the same week that he left here, they got right back to us and said 'Oh no, we're coming back and this is the first thing on the docket and he's doing it.'
"I thought 'Good for him, this is great' because it came from such a pure place.
"I just wanted to give back to the people of the UK that have given me so much for almost 40 years."
The start of Unbroken's music video captures a conversation between the duke and Bon Jovi.
Harry says: "With military people, there's a way of being able to talk about the scars of war in a positive way.
"Some suffer from post-traumatic stress, some of them suffer from depression, anxiety, but they are stronger for it.
"Those are the kind of ambassadors and role models that we need."
Bon Jovi replies: "Well, I thank you for what you have done to bring a light to that. It's what you are doing, really. I just got inspired to write a song from it.
"It is touching, their desire to serve, what they gave, and what they get out of singing."
Unbroken featuring the Invictus Games Choir is out now and available to stream, with proceeds going to the Invictus Games Foundation.