Pardew was given a seven-match ban and £160,000 total fine after admitting an FA misconduct charge relating to his touchline clash with Hull midfielder David Meyler.
The Newcastle boss revealed on Thursday he had written to apologise to Bruce, who is keen to draw a line under the incident.
Bruce said: "As far as we're concerned we accept his apology - all of us in the heat of the moment could look back at times and think, 'what the hell have I done that for?'
"I'm sure Alan will look at himself and think exactly the same thing. In the cold light of day he's been punished heavily and I don't think he'll be going to the races this week.
"I hope it's over - it certainly is as far as we're concerned and I'm sure Alan will make sure it doesn't happen again."