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."