The Everton boss was on Thursday handed an £8,000 fine after accepting a Football Association charge of improper conduct for his actions at the final whistle of the 3-3 draw with Manchester United last weekend.
Moyes and assistant Steve Round, who has also admitted the charge but submitted a plea in mitigation against the fine, ran onto the pitch to confront referee Martin Atkinson after he blew the final whistle with Phil Jagielka closing in on goal.
Everton had scored twice in injury time to level matters and Moyes was incensed Atkinson had not added on more time, describing it as a "terrible decision".
Moyes said: "I was wrong. I shouldn't have run on the pitch. One of the things the referees have asked for is respect. They all wear a logo on their sleeves, the clubs do as well.
"But if they want that from the managers they have to earn it. You have to get that respect by picking up the phone and apologising at the right moments and if you do that then you start to gain respect."
Moyes, meanwhile, has become the latest Premier League manager to disagree with Arsenal boss Arsene Wenger's latest comments about tackling.
The Frenchman criticised Bolton's approach to last weekend's match at the Emirates Stadium, claiming Paul Robinson should have been sent off for a challenge on Abou Diaby.
Bolton boss Owen Coyle and Blackburn's Sam Allardyce, who has had a number of run-ins with Wenger, were quick to dispute the remarks, and Moyes does not believe there is a problem.
He said: "Football's played in many ways and you have to do what you can. I think everybody who plays the game plays within the rules. We certainly do and I don't know of any other managers who don't play within the rules.
"Tackling is a big part of the sport. We mustn't let tackling get taken out of the game. We have to recognise a bad tackle but I'd be more interested in eradicating diving than tackling, I think that's what we should be concentrating on."
Everton continue their search for a first league victory of the season against Newcastle at Goodison Park on Saturday. The Toffees are currently second bottom of the table having managed only two points from their four matches.
Scoring goals had been their big problem until last week, when ironically Moyes' starting XI did not feature any recognised strikers, and the Scot believes they have been unfortunate not to pick up more points.
"We've been playing well," said Moyes. "The players have been doing everything I've asked of them except sticking the ball in the net.
"We need to tidy up a little bit defensively, one or two mistakes we need to sort out, but we've got some good players and hopefully they'll show that."
Newcastle have so far been inconsistent on their return to the top flight, with their two home games resulting in a 6-0 victory over Aston Villa and a 2-0 defeat by Blackpool.
Moyes continued: "The Premier League's like that, throwing up results you aren't sure about and you wonder where they've come from at times.
"But that's why it's such a great league and why every game's so competitive. Newcastle have had a couple of really good results this season and are a big football club."