Eidevall went as far as to call Taylor's comments "borderline slanderous" after their recent encounter in the Women's Super League.
Arsenal emerged 2-1 winners over City on Sunday when Stina Blackstenius struck late to take all three points.
But Taylor was less than pleased with how the Gunners coach handled himself, insisting that he "bullied" the fourth official during the contest.
Eidevall has since hit back at his managerial counterpart, speaking ahead of Arsenal's clash against Bristol City in the Continental Cup on Thursday.
"I was really disappointed with the comments," he said. "There is no truth in them.
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"The wording that Gareth used afterwards is borderline slander. Either he is not very good with the truth or he is not very good with the language.
"I had a good professional dialogue with the officials during the game, I have not spoken to Gareth.
"I don't think his comments are acceptable and I would expect him to reach out with an apology but I am not sure this is going to happen.
"I have no intention of going into any legal process or anything. I have no intention about this getting more of my energy than needed but I really do think it is unacceptable to say things about a fellow colleague that is not true.
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"I think everyone who is at the game and can see my behaviour can also see that is not a single percentage of truth in that and I think that is quite sad to be honest. That's why I think it is unacceptable."
Speaking on Sunday in his post-match press conference, Taylor made the accusation after City lost their unbeaten start to the season.
"That is always the same with him because he is constantly at the fourth official and I think it is bullying," the City coach said.
"I am protecting the fourth official but that's not my job. I can just see that he gets away with it all the time."
The 2-1 for Arsenal saw them jump level with City on 10 points after five games.
Tottenham and Liverpool are both on the same mark, with leaders Chelsea just ahead on 13.