The Wales defender was sent off in Friday's 0-0 draw with the Republic of Ireland in Dublin after a challenge which broke Coleman's leg in two places.
Taylor could be handed an extended ban by FIFA but Press Association Sport understands he has reached out to Everton full-back Coleman to express his remorse over the incident
The Villa left-back is expected to face Norwich in the Sky Bet Championship on Saturday and - after receiving criticism for the challenge - Bruce has backed Taylor.
Bruce said: "He's never going to forget about it but he's got to try and get on with it
He's naturally saddened because it's certainly not Neil Taylor
He's recovered from a serious injury himself (a broken ankle in 2012) so he knows what it is
"He's very disappointed with what's happened, it's saddened him as it's saddened all of us
He's just got to knuckle down and get on with it
"He was reckless, he understands that
He hesitated and was in two minds whether he should go for the challenge but he's caught him
It's part and parcel, we see it every week and players get injured, although none of us want to see it.
"One of the great players in the world (Gareth Bale) could have had a red card one minute before it but thankfully it wasn't as bad on John O'Shea."
FIFA's hearing into the incident is likely to be held within the next fortnight and Taylor's ban could be increased to three games once the world governing body studies the report of match referee Nicola Rizzoli
He is already out of the next World Cup qualifier in Serbia in June but could also be banned for September's qualifying double-header against Austria and Moldova.
Ireland skipper Coleman, who was discharged from St
Vincent's Hospital in Dublin on Wednesday following an operation at the weekend, is expected to be out for at least six months
Bruce added: "The one thing you don't want is to see a fellow professional injured, especially someone like Seamus Coleman who epitomises what's good in a footballer
He's been a terrific professional all these years
From all of us we wish him a speedy recovery."
Source : PA