Baby Archie, who was born at 5.26am on Monday, is also shown in an image alongside his grandmother Doria Ragland and parents Harry and Meghan - who has described her first few days of parenting as "magic".
Could Prince Harry, one of the world's most famous redheads, have named his son after one of the most famous red-haired comic book characters?
The Invictus Games is Prince Harry's other baby of sorts, since he founded for wounded or sick veterans and armed services personnel to compete in a sporting event.
The Clooneys are part of a circle of celebrity friends including Oprah WInfrey, Serena Williams, Priyanka Chopra and David and Victoria Beckham who attended last year's wedding of Prince Harry and his wife, Meghan, who is now the Duchess of Sussex.
Taking to Twitter, the controversial Good Morning Britain host wrote: "So the royal couple, in a striking blow for diversity & feminism, have given their son one of the most popular white middle-class first names, Archie".
According to CNN, Prince Harry was asked if he would be bringing Archie to the Invictus Games in 2020, but replied, "I've no idea".
"He said [Archie] slept for the first 24 hours like all babies do... and then he woke up", JJ said, mentioning that he could clearly see how happy the Duke of Sussex was.
"[The Duke and Duchess of Sussex] are so incredibly grateful for the warm wishes and support they've received from everyone around the world, since welcoming their son two days ago", another caption read.
"He has the sweetest temperament". Noting that it was "changing every single day", he joked that his son "already got a little bit of facial hair".