Raftery also claims another managing director described her as being "knocked up" after she fell pregnant, and that her pay was cut several times after she returned from maternity leave.
And on one occasion, she was allegedly offered a "promotion" – that would involve a US$75,000 ($113,178) pay cut.
She claims she made official complaints against Moore that were effectively ignored by the company.
Other comments allegedly made by Moore to Raftery include: "I don't know what you do all day", "You're not efficient" and "Get the f**k out of my office".
Before the legal documents were filed, Raftery contacted the bank's board to ask that it voluntarily waive its forced arbitration agreement, which would allow her to file her claims in court and have access to a trial by a jury, but her request was rejected.
"Unfortunately, First Republic has refused to waive its forced arbitration provision and seeks to deny Raftery her day in court," Wigdor LLP partner Michael J Willemin said in a statement.
"Apparently the bank has decided that it is more important to protect itself and the men who make money for the bank than Raftery and other women who are victims of unlawful discrimination and harassment.
"We hope that First Republic comes to its senses and changes its mind."