Wade Michael Page, 40, was shot to death by police responding to the Sunday morning attack in the Milwaukee suburb of Oak Creek, the community's chief of police told reporters.
It was the latest violence against the Sikh community in the United States in apparent misdirected revenge for the September 11, 2001, terrorist attacks.
Sunday's attack occurred 16 days after a gunman killed 12 people and wounded scores in a Colorado movie theater, reigniting the gun-control debate in the United States.
"These kinds of terrible and tragic events are happening with too much regularity for us not do some soul-searching and examine additional ways that we can prevent" such violence, President Barack Obama told reporters when asked about the Wisconsin shooting at a White House bill-signing ceremony.
However, New York Mayor Michael Bloomberg criticized Obama and certain Republican presidential nominee Mitt Romney for not advocating tougher laws to prevent dangerous people from obtaining guns.
"The fact that criminals, terrorists and other mentally ill people have access to guns is a national crisis," Bloomberg said during a visit to a Sikh community center in Queens.
Bernard Zapor, the Bureau of Alcohol, Tobacco, Firearms and Explosives special agent in the investigation, said Monday that the 9 mm semiautomatic handgun with multiple ammunition magazines used by the attacker had been legally purchased.
The gunman shot people inside and outside the Sikh house of worship, including a police officer, Oak Creek Police Chief John Edwards said. Another police officer with a rifle then shot the gunman, who died at the scene.
While authorities said Page was the only gunman, they also had been looking for another man spotted at the crime scene Sunday who left before anyone could identify him. On Monday, the FBI's Paul Bresson said the man being sought had been "located, interviewed and cleared."
According to Edwards and the FBI, authorities have received tips that Page might have links to the white supremacist movement, but nothing had been confirmed.
Two neighbors of Page identified him in photos that showed him playing in the far-right punk band "End Apathy," and the nephew of the slain president of the Sikh temple said the attacker had a 9/11 tattoo on his arm.
Teresa Carlson, the FBI special agent in charge of the investigation of Sunday's shooting, said no motive for the attack has been established. The FBI was looking into whether it was domestic terrorism, which is the use of violence for political or social gain, Carlson said.
"We are looking at ties to white supremacist groups," Carlson told a news conference, adding there was no active investigation of Page prior to Sunday's attack.
According to a man who described himself as an old Army buddy of Page's, the attacker talked about "racial holy war" when they served together in the 1990s. Christopher Robillard of Oregon, who said he lost contact with Page more than a decade ago, added that when Page would rant, "it would be about mostly any non-white person."
"He didn't seem like the type of person to go out and hurt people," Robillard said. "He would talk about it all the time, but it was more like he was waiting for the ... revolution to start."
Page, born on Veterans Day in 1971, joined the Army in 1992 and left the service in 1998, according to Army spokesman George Wright. His service was marked by "patterns of misconduct," though he received an honorable discharge, according to a Pentagon official.
The suspect did have a criminal record, Edwards said. A background check showed Page had separate convictions for DUI in Colorado in 1999 and for criminal mischief in Texas in 1994.
A federal law enforcement official told CNN that investigators interviewed a former girlfriend of Page. The woman, who had recently broken up with him, told investigators she had no indication Page planned such an attack. She provided the names of friends and associates of Page, the official said.
Because of their customary beards and turbans, Sikh men are often confused with Muslims, and they have been the targets of hate crimes since 9/11 terrorist attacks on New York and Washington.