The controversial "Wrecking Ball" singer posted a message on Twitter last night that appeared to poke fun at O'Connor's illness.
The pop star tweeted: "Before Amanda Bynes…There was…," with a picture of some of O'Connor's previous tweets attached.
O'Connor's past tweets included: “I desperately need to get back on meds today. Am in serious danger," and, "Does anyone know a psychiatrist in Dublin or Wicklow who could urgently see me today please? I'm really un-well… and in danger."
Paul Jenkins, CEO of charity Rethink Mental Illness, said: "Miley's tweet appears to be an attempt to discredit Sinead by highlighting her mental health problems.
"If that is the case, it is a cruel and unkind response. Sinead O'Connor has helped break down the stigma surrounding mental illness by speaking openly about her own struggles.
"That takes a lot of bravery, so to use it as ammunition against her, if that is the case, would be deeply insensitive. We have asked Miley to explain what she meant by her tweet and we urge her to clarify what she wanted to say."
Mental health charity SANE also commented on the exchange between the two singers on social media, which started yesterday when O'Connor wrote an open letter to Cyrus warning her that the music industry would "prostitute" her.
Marjorie Wallace, chief executive of SANE, said: "This sad and distasteful exchange between two talented singers would never have taken place so publicly had it not been for social media. At SANE, we have always warned of the potential psychological damage caused by the abuse of social media, especially when people are fragile."
She continued: "Miley Cyrus may have a powerful voice and sings a powerful song, but her ignorance about mental health could do untold damage, not only to Sinead O'Connor but to the thousands of people who have suffered a mental breakdown or enduring illness.
"It is precisely this mockery and lack of sensitivity which reinforces the stigma of mental illness, leaving people isolated and afraid and, in some cases, thinking of taking their own lives. It is irresponsible and she should apologise."