Sent off in the FA Cup defeat to Brighton last week, Gordon is now suspended for next week's Carabao Cup final against Liverpool.
Ahead of Monday night's clash with West Ham, Howe stated: "We discussed it as a club, with the respected people that we need to discuss it with, lawyers etc.
"We just felt that there probably wasn't sufficient grounds to do that.
"There was no intention from Anthony. Without going into too much detail on the actual event itself, I think the player (Jan Paul van Hecke) sort of bends down, and it makes the push look worse.
"There's no intention from him to hurt anybody. But you have to look at it and go 'does this have a chance of being successful?' - if there's no chance of being successful, what's the point of appealing, so Anthony understands."
Howe admits Gordon is hugely disappointed, as the manager is, given the form and impact the winger has made in recent weeks.
He continued: "I think for Anthony of course he was hugely disappointed on Sunday.
"I think there's that realisation once - because your emotions are in the game that you're playing & it can't be for the future - the game ends, you sort of realise what's happened and the repercussions of that incident.
"So I'm sure Sunday night at home, or on Monday, he would have been feeling all the potential negative emotions of that. But I think he's very quickly come back to us, trained really well, you wouldn't know there's a disappointment there.
"Huge respect to him for that, and he's trying to help his teammates because now he knows his job is to train really well, be there as a support for his teammates and try and push us on to achieve something great in his absence."
Meanwhile, Howe also discussed the season-ending injury suffered by wing-back Lewis Hall. It was decided that Hall should undergo ankle surgery rather than play through the pain for the remainder of the campaign.
The manager said, "Yeah, really disappointing for us. I'm devastated for him actually because he had so much to look forward to at this moment of the season with the cup final - which will be a big thing for all the players at the moment.
"Also the international experience coming up, and he was just in really good form as well and playing very well, so yeah, that's a big blow for him and for us.
"The most important thing now is his recovery and making sure he comes back as good as he can."
Quizzed for more detail on the injury, Howe continued: "It was from the game (defeat at Liverpool) but I I don't necessarily think it was in that game.
"I think there was a slight build upon speaking to Lewis afterwards and trying to find out the sort of timeline of the injury - but that was the game where I think the feeling in his ankle had just got to the point where it needed investigating.
"He played very well in the game, but he just had a bit of stiffness and swelling afterwards. So we investigated and obviously it's landed at this point."
Howe was also keen to talk up midfielder Joelinton's call-up to the Brazil squad. The selection a great endorsement for both player and club.
Howe said: "Huge news for him. We're delighted for Joe. He's such a passionate guy.
"He's had his own injury troubles and I thought one of the highs of Sunday was to see him return.
"Not just return but return in his best way; running, chasing, winning balls, being a reinforcer for us in that position. So I'm delighted for him and I think it's just testament really to his character."