John Constantino, 64, of Mount Laurel, N.J., poured the contents of a canister of gasoline on himself in the center portion of the mall Friday afternoon, police said.
He then set himself ablaze, with passing joggers taking off their shirts to help put out the flames.
Police had said Constantino was conscious and breathing at the scene, but he died later that night at a Washington hospital.
"John Constantino was a loving father and husband. His death was not a political act or statement, but the result of his long battle with mental illness," his family said in a statement issued through attorney Jeffrey Cox.
The statement did not address the nature of the mental illness.
The family said in the statement that it "would like to acknowledge the heroism of the paramedics and bystanders who attempted to save" Constantino.
The family "is shocked and deeply saddened," the statement said. Describing the incident as a "personal family matter and not an issue of public concern," it asked that the media respect its request for privacy.
The self-immolation on Friday occurred in public view, on a central national gathering place, in a city still rattled by a mass shooting last month and a high-speed car chase outside the U.S. Capitol on Thursday that ended with a woman being shot dead by police.
Constantino was standing by himself in the center portion of the Mall when he emptied the contents of a red gasoline can on himself and set himself on fire moments later, said Katy Scheflen, who witnessed it as she walked across the area.
Police say they responded around 4:20 p.m. Friday.
Scheflen said passing joggers took off their shirts in an effort to help douse the flames, and the man was clearly alive as the fire spread. A police department spokesman said he was conscious and breathing at the scene. MedStar Washington Hospital Center tweeted that the man was taken there and he was in critical condition.
"There was not a lot people could do because it was a gasoline fire," Scheflen said.
She said he may have said something before he acted "but it was nothing intelligible." She said she did not see him holding any signs before he set himself ablaze and that there was another man with a tripod set up near him but did not mention if he was filming.
Lt. Pamela Smith of the U.S. Park Police, which is investigating along with the D.C. police department, said: "I'm not aware of any signage or any articulation of any causes."
Eric Tucker of the Associated Press contributed to this report.