Shadow home secretary Ms Abbott retweeted a series of critical remarks about the deputy leader, including one questioning his position.
Labour MPs Emma Dent Coad, Danielle Rowley and Ms Abbott were among those condemning Mr Watson, who wrote a critical letter to National Executive Committee (NEC) general secretary Jennie Formby.
Ms Abbott shared one tweet from an NEC representative criticising Mr Watson’s treatment of Ms Formby.
“This is not behaviour befitting of the Office of Deputy Leader. You should consider your position?” Claudia Webbe’s tweet added.
Other tweets promoted by Ms Abbott included ones accusing Mr Watson of “appalling behaviour” and of undermining staff.
Ms Dent Coad expressing “solidarity” with Ms Formby, who was heavily criticised alongside Labour’s director of communications Seumas Milne in the BBC report for allegedly interfering with anti-Semitism investigations.
“On the day a report on parliamentary bullying is published, Watson has been relentless. Shame on him,” the MP tweeted.
Ms Rowley said it was “disgraceful” for Mr Watson to unnecessarily “attack” Ms Formby publicly at a time when she is undergoing chemotherapy for breast cancer.
Mr Watson had written to the NEC general secretary calling for the party to publish its submission to a watchdog investigating anti-Semitism claims and raising allegations that she had deleted emails relating to cases, which Ms Formby denies.
His letter came after eight former Labour officials told Panorama they had been undermined in their attempts to tackle the blight in the party.
Former Labour general secretary Lord Iain McNicol and three others broke non-disclosure agreements (NDAs) by speaking out.
Labour has denied the claims and written a complaint to the BBC.
Mr Watson’s letter said those who spoke to the programme were “very brave to go before a camera and tell their stories”.
“The way that they have been smeared, including by Labour spokespeople, is deplorable,” he said.
“Even if some in the party did not want to hear what they had to say, it is unacceptable to attempt to undermine their integrity and characters in this manner.”