Kirsty Blackman, the SNP MP for Aberdeen North, retweeted a post calling for the Edinburgh South West MP to be expelled for the party to allow the SNP to live up to its “zero tolerance” stance on trans rights.
Ms Cherry and Ms Blackman have been long-term opponents on the issue of self-identification for transgender individuals and have been at the heart of divisions within the party over the issue.
Asked about the row on BBC’s Good Morning Scotland on Monday, deputy first minister John Swinney called for “considered and respectful debate”.
It also comes months after Ms Cherry accused Ms Blackman of “lies and smears”.
The row erupted after Ms Blackman retweeted a post, which stated: “ If my party truly stands for trans rights and equality, if our stance is truly 'zero tolerance', then it has to start from within. Joanna C must be expelled from the SNP.
"Show the people of Scotland and the rest of the UK that 'zero tolerance' means exactly that."
The tweet was in response to a post from Ms Cherry, who had responded to another Twitter user who had claimed Nicola Sturgeon wants to “let lads into women’s spaces” in a post directed to Neil Gray, an SNP MSP.
Responding, Ms Cherry posted: “Thank you @ellegla. Misogyny and abuse of women is never ok even when they have been impertinent enough to disagree with the new orthodoxy of queer theory and stick up for their rights. Tackling this problem should start at home."
The high-profile MP was sacked from the SNP’s frontbench in February as the Alex Salmond affair divided loyalties among members.
These divisions within the SNP partially led to the creation of the new pro-independence party, Alba, led by the former first minister, Mr Salmond, which has a gender critical stance on trans rights.
Ms Cherry, posting this morning by retweeting Mr Swinney’s response to the row, reiterated her opposition to self-identification.
She said: “As a lesbian and a feminist I’ve spent a lifetime campaigning for equality and, to be clear, I support trans rights. What I don’t support is the right of any man to self ID as a woman and access the single sex spaces which the Equality Act protects for women and girls
“The Equality Act also makes it unlawful to discriminate against anyone, harass them or victimise them on the grounds of their sex, sexual orientation or beliefs and this law applies to political parties & their representatives as membership organisations.”
Ms Blackman later deleted the retweet, stating: “Have deleted an earlier RT. I am clear though, the SNP needs to do more to tackle internal transphobia, including sanctioning or expelling those in the party who are transphobic.”
In January, Ms Sturgeon took to Twitter to make a plea to young activists after reports of “significant numbers” leaving the party due to concerns around trans rights.
In a scripted video message, the First Minister and party leader said she was concerned that young activists did not view the SNP as having a strong enough stance against transphobia, adding that it should be treated with “zero tolerance”.