Kenny Williams also said that his son was playing with an injured left shoulder Sunday, an injury that happened in the third quarter. Kenny Williams added that his family does not make excuses and did not blame his son's miscues on the injury.
Injury or not, some people were apparently outraged enough to wish harm to Kyle Williams after his two turnovers, the second of which led directly to the Giants' game-winning field goal.
One early threat, via Twitter, wished harm on Kyle Williams' wife and child. He is single with no children.
"I'm used to the years of criticism and threats on my life from time to time, but I have to hear about threats on your son's life while you're watching TV and it certainly makes you question our culture of sports as it stands," Kenny Williams said.
For his part, Kyle Williams said Monday he takes full responsibility for the fumble, calling the moment "painful." And Williams' teammates are on his side, too, along with others from around the NFL who have called in support after San Francisco's season ended.
He insisted it's part of his job to face the criticism in the aftermath of his mistakes and not hide out in a difficult time. Perhaps that will go a little way in calming down the angry fans, some of whom wished harm on Williams and his family after the gaffes.
"It's one of those things you have to take accountability for," Williams said. "Everybody is responsible for what they do on the field. It's something that I was responsible for and I made a mistake and it's time to own up to it and move forward."
Kicker David Akers took to Twitter on Monday asking people to stop with the harsh remarks directed toward his teammate, saying "ppl need to get a grip! Leave Kyle alone!" He also reminded everybody the NFC West-champion 49ers (14-4) win and lose together, so there would be no finger-pointing in the locker room.
"I'm irritated with the way people are treating him, absolutely," Akers said. "I think it's ridiculous. You know, get a grip on what life's about. He went out and he put his soul out there. That's what he does. He was not trying to do anything other than make an incredible play for this team. He had a great kickoff return. Mistakes happen. We all make mistakes. But when you're out there truly battling to do the best you can, my hat goes off to him, to anybody that does that."
49ers president and CEO Jed York also chose Twitter to support Williams, writing: "Keep your head up, @KyleWilliams_10. We win and lose as a team. With Class."
A year ago with the Eagles, Akers missed a pair of field goals in the NFC divisional playoffs as Philadelphia lost 21-16 to the eventual Super Bowl champion Green Bay Packers.
"I was that guy last year," Akers said. "It's tough because we're all competitors and we all give it everything we have. Kyle's made a lot of big plays for us. He's just trying to make plays out there. The weather conditions were horrible. I know he'd never give any excuses for any of that. I say this and I say it in the truest way: We win as a team, we lose as a team."
Kenny Williams broached the subject of the death threats on his own after he was asked if Sunday was the toughest day he had ever experienced in sports. He said it was by a long shot.
"I told his mother on the way out exactly what was coming," Kenny Williams said about the threats. "That's the nature of the beast."
Kenny Williams said that NFL security swiftly moved into action and that all threats were being taken seriously.
He was asked if he was disappointed that sports fans would go to the extreme of making such threats.
"It doesn't matter what I think, I can't change it," Kenny Williams said. "The only way to change it is to go into another occupation if you can't deal with it."
Kenny Williams said that he not only thinks his son can handle the adversity from Sunday's game, but that it will help him to thrive in the long run. First things first, though, is to address the injury.
Kenny Williams confirmed that Kyle Williams was undergoing treatment Monday with the 49ers training staff for an unspecified shoulder injury.
"He thinks he may have separated his shoulder in the third quarter so he is in the training room," said Kenny Williams. "That information he has not said anything about publicly and will not because he felt strongly that he could get the job done. There are no excuses; we don't do that. I don't want that misconstrued."
In his season-ending news conference Monday, 49ers coach Jim Harbaugh confirmed that Kyle Williams was examined for a shoulder injury.
"There's no separations, but he's very sore," Harbaugh said. "He told me that today. He was soldiering through it. I wasn't aware of it during the game."
Kyle Williams, a second-year player out of Arizona State, not only had the fumble in overtime, but early in the fourth quarter a rolling punt glanced off his knee. The Giants recovered that at the 49ers' 29-yard line and scored a go-ahead touchdown less than three minutes later.
Asked if the shoulder injury played a role in either of his son's miscues, Kenny Williams was blunt.
"He felt he could get the job done. He didn't. End of story," Kenny Williams said.
Doug Padilla covers the White Sox for ESPNChicago.com and ESPN 1000. Information from The Associated Press contributed to this report.