Clooney, 57, said history was repeating itself in the case of Meghan and Diana, warning “we’ve seen how that ends”.
His intervention comes amid reports of a rift between Meghan, who is pregnant, and her father, Thomas Markle.
Over the weekend a newspaper published a deeply personal letter allegedly written by the duchess to her father in August, months after he missed her wedding.
After joking about a report suggesting he would be named godfather to Harry and Meghan’s baby, Clooney, who was speaking at an event in Los Angeles, said: “I do want to say, I just saw this piece.
“They’re just chasing Meghan Markle everywhere, she’s been pursued and vilified.
“She’s a woman who is seven months pregnant and she is being pursued and vilified and chased in the same way that Diana was, and it’s history repeating itself.
“And we’ve seen how that ends.
“I can’t tell you how frustrating it is to see that.
“You’re taking a letter from a daughter to a father and broadcasting it everywhere.
“She’s getting a raw deal there, it’s irresponsible.
“I’m sort of surprised by that.”
Clooney and his wife have 18-month-old twins, daughter, Ella, and a son, Alexander.
Asked if he was going to be godfather to the royal baby, he said: “No, I’m now a father of twins, I’ve got enough shit. Literally, literally shit.”
Former Suits actress Meghan, 37, is estranged from her father after a string of high-profile incidents since she became engaged to Harry.
Days before his daughter was due to marry into the royal family, Mr Markle allegedly staged paparazzi photos of himself.
He has previously said his relationship with his daughter has broken down and has made public appeals for her to get in touch with him, including an appearance on Good Morning Britain.
Mr Markle, speaking to the Mail On Sunday this week, dismissed claims he has not attempted to contact his daughter.
An unnamed friend of Los Angeles-born Meghan told US magazine People: “He’s never called; he’s never texted. It’s super painful.”
Five nameless friends of the duchess spoke to People in an effort to counter the “global bullying” directed at her.
Meghan revealed in January that her baby is due around the end of April or beginning of May.