Race and gender are "absolutely" a factor, Clinton said during an interview with the BBC on Tuesday.
Meghan Markle recently said she struggled to deal with the repercussions of being a new mother in the spotlight and together with her husband is suing a British newspaper for releasing an edited version of a private letter she sent to her father.
Clinton said "race was clearly an element" in some of the comments Meghan has received on social media.
"To think that some of your -- what we would call mainstream -- media actually allowed that to be printed in their pages, or amplified, was heartbreaking and wrong," Clinton said in a joint interview on BBC Radio 5 Live alongside her daughter Chelsea.
"I feel as a mother I just want to put my arms around her.
"Oh my God, I want to hug her. I want to tell her to hang in there, don't let those bad guys get you down, keep going, do what you think is right."
The former first lady was full of praise for the Duchess, who she called "an amazing young woman."
"She has stood up for herself, she has made her own way in the world," said Clinton.
"And then she falls in love, and he falls in love with her, and everybody should be celebrating that because it is a true love story."
Clinton said that Meghan could benefit from learning some "techniques" like "some humor, some deflection," but added: "It is tough what she is going through and I think she deserves a lot better."
Meghan gave a rare glimpse into her personal life in an interview with ITV in October, talking about the issues she's grappled with as a new wife, mother, and duchess whose life has been judged and dissected by the media and public since she first began dating Prince Harry.
"Not many people have asked if I'm OK, but it's a very real thing to be going through behind the scenes," she said.
Clinton is in the UK as part of a book tour with her daughter.