Jessica Cruz, executive director of the charity, accused Menzel of "joking about self-harm when self-harm is not a joke".
She added: "Self-harm and suicide is a very serious situation. Some people may not realise the impact of that type of statement.
"It shows a lack of education around the pain families and individuals go through and further stigmatises mental health."
The singer, who is perhaps best known for voicing the role of Elsa in the Disney film, Frozen, was on the show to promote her role in a television remake of the iconic 1988 film, Beaches. She has taken on the role of CC Bloom, which was initially played by fellow Jewish actress, Bette Midler.