Speaking with ET's Nancy O'Dell at Hulu's For Your Consideration event for his new show, Catch-22, the Oscar winner was asked how he felt that they didn't include the name George in their baby's moniker at all.
"That kind of hurts," Clooney deadpanned, before laughing and admitted, "I like it! Archie. It's good!"
When it was brought to his attention that it's possible (although very unlikely) that they may have turned to actor Harrison Ford for their baby's middle name, the star said that he understood the choice.
"Well, he was president in a movie, once," Clooney reasoned with a shrug, before admitting his own reverence for the Star Wars icon. "And, if you think about it, it is Harrison Ford. I'm gonna [re]name my daughter Harrison now."
In reality, many believe the name Harrison is a play on the patronymic tradition of naming a child after their father, as in "Harry's son." But, with no official confirmation, it could still be a reference to the Han Solo actor.
ET spoke with Clooney on Tuesday at the premiere of Catch-22 in Hollywood, one day after the royal baby's birth, which happened to also be Clooney's 58th birthday, and the actor joked that the new bundle of joy "is stealing my thunder!"
"This was my day! I was sharing it already with Orson Welles and Sigmund Freud!" Clooney quipped.
While Clooney feigned disappointment over sharing the b-day attention, the actor -- who was one of the celeb guests as the Royal wedding when the Duke and Duchess of Sussex tied the knot almost a year ago -- said he was overjoyed for the new parents.
"I'm very happy. They're a lovely couple, so it's very exciting," Clooney marveled.
However, despite having two children of his own -- 1-year-old twins Alex and Ella, whom he shares with wife Amal Clooney -- Clooney said that he doesn't feel it's his place to give the Duke and Duchess, or anyone else for that matter, advice on anything.
"I don't give advice, I'm 58," he said. "At 58 you just go, 'Ugggh.' I just walk with a walker everywhere I go."
Check out the video below to hear more.
Clooney's new series, Catch-22, premieres on Hulu on May 17.