"I share and fully support the league's concerns and goals on player safety," the New Orleans coach said in a written statement released by the team. "It is, and should be paramount.
"Respecting our great game and the NFL shield is extremely important to me," Payton added, referring to the league's famous logo.
NFL Commissioner Roger Goodell on Wednesday suspended Payton for the 2012 season, effective April 1, one of several unprecedented penalties he issued against the Saints.
Payton said that, as head coach, he should take full responsibility for an operation which the NFL says offered improper cash bonuses for blows that either knocked targeted star players out of the game or left them needing help off the field.
The NFL has said Payton initially lied to NFL investigators about the program, at first denying its existence, and also instructed his defensive assistants to lie. The league also slapped an eight-game suspension on general manager Mickey Loomis and a six-game suspension on assistant head coach Joe Vitt, who also coaches linebackers.
The NFL has said the bounty program started in 2009, Williams' first season, when the Saints also won their only Super Bowl. The league also found that the program continued through 2011 even after NFL officials told the Saints to check into it and put a stop to it if they found anything improper was going on.
Payton issued one earlier written apology a few days after the NFL first released the findings of its probe on March 2.
In his latest statement, Payton said the Saints "will implement all necessary protections and protocols, and I will be more vigilant going forward."
Payton, however, won't begin exercising that vigilance until after New Orleans hosts the Super Bowl next February, when his suspension officially ends.
The coach thanked team owner Tom Benson, his players and Saints fans "for their overwhelming support."
Some Saints players have already been in touch with Payton to let him know they are behind him, and starting right tackle Zach Strief said on Friday players "owe it" to their embattled coach to come closer together, live up to the standards he set during the past six seasons and do everything they can to win in his honor while he is away.
"This season is going to take on an us-versus-the-world mentality," Strief said. "The perception at this point is not positive with us. We've worked very hard, I know as players, to gain respect, not only as a good team but as a good group of guys. That perception will go completely out the window now, and I think that'll be kind of a rallying cry for us. It will give us a chance to kind of come together closer than we've ever been and say, `Look, nobody wants us to even win anymore, and we're going to win for each other, and for coach Payton and Mickey and the guys that essentially took this fall."'
The NFL has said punishment for players who participated in the bounty program will come later, meaning the Saints could potentially see some defensive regulars suspended as well. The league has said between 22 and 27 players participated in the program, which usually paid hundreds of dollars - a symbolic gesture relative to the size of NFL contracts - but sometimes thousands.
The payments were made out of a pool funded by players, coaches and others. Payouts were also awarded for big plays such as forced fumbles, sacks and interceptions, but such compensation is not allowed under the NFL's collective bargaining agreement. The NFL said the pool grew as large as $50,000 during the Saints championship season.
Strief noted that the Saints already had a sampling of what it was like to operate with Payton being less present in the weeks after he broke his leg last season.
"He couldn't get off a (trainer's) table for like three weeks. He wasn't in meetings. He wasn't at practice," Strief said. "There's a tremendous amount of faith in the offensive staff. Even after he came back, I think he really kind of gave a lot more responsibility to the guys that had been doing it. He really kind of said, `All right, you've been doing it, you're doing a good job, go ahead and keep going."'
"Our offensive staff last year really worked very much more independently than they ever had before," Strief said. "So I think in that respect, we're somewhat used to that already."
Strief said Saints veterans have been with Payton long enough now that they know how he likes the structure and tempo of practice, and that they even worked to simulate that during last year's lockout, when Brees gathered players together for offseason workouts at Tulane.
Strief said he visited Payton on Thursday.
"It crushes you to see him and I know it was unexpected for him," Strief said. "For all of us, really, it was completely unexpected.
"When I was hurt last year, for me to watch a game on TV was miserable. I can only imagine what it's going to be like for him," said Strief, who sprained his right knee and missed part of the season. "And I know that for me personally, he will be a big motivating factor in working hard and being a better player because he deserves to have us do that."