Eta has been named interim coach for the last five games of the season as Union aims to avoid relegation from Germany's Bundesliga.
Union Berlin condemned sexist comments on social media after appointing Marie-Louise Eta as the first female head coach in Europe's "big five" football leagues, pledging the club "has her back".
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After the announcement of Eta's appointment on Sunday, a series of critical comments on social media targeted her, claiming she was unsuitable for the job because she is a woman.
"With all due respect, that's sexism," Union's account on X responded to a post arguing that players would not take a woman's tactical instructions seriously.
The club also listed another comment stating that a male coach who lost to her would lose face as sexist.
Eta is a longtime member of the club's coaching staff, having served as an assistant to the men's team and as head coach of the under-19 team.
Responding to a comment expressing concern about the treatment of Eta and potential sexist backlash if she loses a game, the club posted: "The Union family has her back."
Eta has been named interim coach for the last five games of the season as Union aims to secure its spot in the Bundesliga for next season.
Union previously said she will take over as head coach of the club's women's team for next season.
The decision to appoint Eta won praise from Berlin's mayor Kai Wegener, who called the decision "a strong signal for professional football and for women in elite-level sports," but spelt Eta's name wrong in the process.
The club corrected him. "We were so overwhelmed," Wegner answered.
Eta's first Bundesliga game as interim manager is on Saturday against Wolfsburg.
Europe's "big five" includes the Bundesliga, England's Premier League, the Spanish La Liga, Serie A and Ligue 1. These are considered the continent's most prestigious domestic competitions.
Women have managed men's teams in other European leagues, but none had previously coached in any of the big five until Eta's appointment.
The club dismissed its previous head coach Steffen Baumgart earlier this month after a series of poor results left Union in the relegation zone.