England forward Stanway was shown a straight red card by referee Rebecca Welch 10 minutes before half-time in the 2-2 draw for her challenge on United winger Leah Galton.
Stanway said on Twitter she was sorry for her dismissal and that there “was absolutely no malice intended” when she made the tackle.
“I’m glad Leah was OK and able to carry on with the game,” Stanway wrote.
“A huge thank you to my Man City team-mates for the support and effort to secure a point. And to both sets of fans for creating an unbelievable atmosphere.
“I made a mistake and hope everyone can accept my apology, but I shouldn’t have to log into social media and find myself subjected to all sorts of abusive messages – remember we are all human and mistakes happen.”
Khadija Shaw gave City a half-time lead and after Lucy Staniforth and Alessia Russo had put United in front, substitute Ellen White tapped home an equaliser.
City ended a run of three straight league defeats, but sit ninth in the Women’s Super League table after winning only one of their first five games, while United were two points behind leaders Arsenal before Sunday’s games.