The Edinburgh MP, who is running to be the next UK deputy leader of his party, has said there should always be a position for a woman at the top of Labour and said more flexible working arrangements should be introduced at Westminster and Holyrood to improve the life balance of MP and MSPs who are parents.
Mr Murray made 12 pledges to the Women's Parliamentary Labour Party, including tackling sexual harassment and bullying in the party - and backing his MP colleague Stella Creasy’s campaign to make misogyny a hate crime.
The party has become bitterly divided over the issue of trans rights, and its policy on allowing trans women to be selected on All Women Shortlists had previously prompted 300 female members to resign.
The issue has reignited in recent weeks, with two of the three Labour leadership candidates signing a trans rights pledge, which encouraged the expulsion of members for "transphobia" and claimed that a new organisation established by some long-standing Labour and trade union members, Women's Place UK, was a "hate group".
And on Monday another new group, Labour Women’s Declaration, is holding a rally in London to call on Labour MPs to support the party's 2019 manifesto commitment to enforcing sex-based rights.
He added: "There is so much more to do. This starts with reforming our own party, which is why I am determined there should always be at least one woman in our leadership team.
“Too often when we talk about gender equality we ask what more can be done to support women to break through - rather than do the harder of work of seeking to change the system. I want to change the system.
“Public service is something to be championed, but we cannot hope to encourage more people to choose it as a career unless we remove institutional barriers.”
Mr Murray's other pledges include calling for formal parental leave arrangements for all levels of elected public office, using proxy voting and locums during maternity leave - including in the Scottish Parliament – and proxy voting options for those with caring responsibilities and for those living in remote constituencies.
His proposals come as Labour MSP Jenny Marra is due to start maternity leave and the SNP's Gail Ross announcing plans to stand down from her rural seat in the far north of Scotland because of the time spent away from her family due to travelling to and from Edinburgh.
Mr Murray said he also backs job-sharing roles, such as the original idea put forward by Jackie Baillie MSP and Pauline McNeill MSP to share the role of Scottish Labour deputy leader, which was rejected by the party.