The feud began after an interview in Rolling Stone magazine in which Cyrus said she was inspired for her "Wrecking Ball" music video by an image of a single tear in O'Connor's "Nothing Compares 2 U" video.
O'Connor responded to Cyrus' comment with an open letter expressing her concern for the younger pop star's recent behavior.
"I am extremely concerned for you that those around you have led you to believe, or encouraged you in your own belief, that it is in any way 'cool' to be naked and licking sledgehammers in your videos," O'Connor wrote in reference to "Wrecking Ball."
Cyrus responded by posting a screenshot of tweets by O'Connor from last year, in which the Irish singer asked for recommendations for a psychiatrist for her bipolar disorder.
Cyrus compared O'Connor to actress Amanda Bynes, who is currently under psychiatric treatment, Sky News reported.
"It is most unbecoming of you to respond in such a fashion to someone who expressed care for you," O'Connor wrote in an open letter to Cyrus. "I hope that you will apologize to Amanda Bynes and to any person who has been wounded by your mockery of those who have suffered."