Gaffney directed profane tweets at the fan after the Redskins lost 27-24 in overtime to the Cowboys on Sunday.
When the fan taunted Gaffney over the Redskins' losing record, Gaffney responded: "3-7 ain't a record to be proud of I'm just proud I ain't you get a life or kill urself."
Gaffney later tweeted that he didn't really want the fan to kill himself.
On Monday, Gaffney said: "It's something that got took out of 'text. Cowboy fans, that's how they are. I was just telling the dude: 'Y'all won. All right, leave me alone. Don't talk to me.' And that's all it was."
The NFL and the Redskins both have talked to Gaffney about his tweet, a source told ESPN NFL Insider Adam Schefter.
Having given his explanation to reporters, Gaffney went back on Twitter and stoked the issue again.
"They say I can't tell people to kill themselves didn't know freedom of speech had limitations," he tweeted.
Coach Mike Shanahan skirted the issue when asked about it at his weekly news conference.
"No offense, but I have no idea of how to even get on Twitter, I really don't," Shanahan said. "I don't even know what he said."
That seemed like an incredible answer coming from a coach who constantly monitors his players and keeps a close rein on everything happening within the organization.
"I heard he said something," Shanahan explained. "But I said, 'Don't tell me what he said until after the press conference, so I don't have to comment on it.' Because I was really getting ready for (the next game at) Seattle, and I don't want to kind of get into detail in that. I'm just trying to honest with you."
Information from The Associated Press was used in this report.