The Invictus Games brings together current and former wounded, injured or sick servicemen and women from more than 20 countries.
In a statement posted on YouTube, the Duke of Sussex said he was "so grateful" to organisers for attempting to find "any alternative to try and carry out the games in a different way - in a safe way".
He added the decision to not hold the event was the "most sensible and the safest option".
The Duke said he was "really excited" about the event to be held in 2021.
Harry also urged people to "reach out" and support "those people in your community who you know may be suffering or may be vulnerable in this period".
In a statement, the Invictus Games Foundation said: "The decision was also taken in recognition of the anticipated strain on medical staff and the infrastructure required in dealing with the pandemic.
"We did not wish to add to the complexity of the response, or increase the risk to those involved, by bringing together an international and potentially vulnerable audience."
It added: "We are now investigating all options to reschedule the Invictus Games, subject to the availability of key facilities and resources, to May or June 2021.
Organiser said planning is already underway, though the Duke of Sussex said in his statement that a date was yet to be confirmed.
A series of Royal engagements also look set to be cancelled or postponed.
The Queen cancelled upcoming visits to Cheshire and Camden and has moved to Windsor early ahead of her Easter break, following Government advice to social distance amid the outbreak.