But the president stopped short of saying he would call for new gun control.
"We're still awaiting the outcome of a full investigation," he told reporters at the White House. "All of us are heartbroken by what happened."
Obama said such attacks happen with "too much regularity."
The alleged shooter in the deadly attack at a Sikh temple was an Army veteran and a white supremacist, officials said Monday.
Wade Michael Page, 40, was identified as the person who fatally shot six people and wounded three others before being shot and killed by a police officer at the Sikh Temple in Oak Creek Sunday, the Milwaukee Journal Sentinel reported.
The Southern Poverty Law Center, a group that has studied hate crimes for decades, said Monday there was "no question" Page, a former member of the U.S. military, was a white supremacist and that her group had been tracking him since 2000.
"He was involved in the scene," said Heidi Beirich, director of the center's intelligence project, adding that Page was the leader of a racist white-power band called End Apathy.
Beirich said her group began tracking Page after he tried to buy goods from the National Alliance, a well-known hate group.
However, federal law enforcement officials told NBC News the unidentified suspect had no obvious connection to domestic terror or white-supremacist groups and apparently was not on any list of suspected terrorists.
Oak Creek Police Chief John Edwards earlier said he believed the shooting was domestic terrorism -- a designation that implies a political or social agenda.
The Los Angeles Times reported tattoos on the dead suspect and certain biographical details contributed to a possible domestic-terrorism scenario.
"My focus is not on what category it is but what happened and the loss of life," U.S. Attorney James Santelle, who is personally reviewing the Sikh temple massacre, told the Journal Sentinel.
The Journal Sentinel and WTMJ-TV, Milwaukee, reported the shooter was a white male in his 40s, and WTMJ said he had an arrest record for minor offenses.
The Journal Sentinel cited a source who said one firearm was recovered as well as multiple magazines, or ammunition storage and feeding devices. WTMJ said police recovered at least two semiautomatic handguns.
The FBI, which is leading the investigation, investigated a home in Cudahy, Wis., 7 miles northeast of Oak Creek and 2 miles from the Milwaukee border, in relation to the shooting.
Kurt Weins said he rented the property to a man he believed to be from Chicago, the Journal Sentinel reported.
Weins said police told him not to disclose the name of his tenant. He said he didn't find the man's behavior to be suspicious, though he was a bit of a loner.
"I had him checked out and he definitely checked out," Weins said.
Witnesses said the gunman stalked through the 17,000-square-foot temple about 10:30 a.m. Congregants ran for shelter and barricaded themselves in bathrooms, prayer halls and closets, where they made desperate cellphone calls and sent anguished text messages pleading for help.
Those who hid included 20 to 25 women cooking lunch in the basement with five to 10 children, Darshan Dhaliwal, who identified himself as a leader at the temple, told the Journal Sentinel.
The gunman's rampage ended when one of the first police officers to arrive shot and killed him. Another police officer who tried to aid a victim was ambushed by the gunman and shot multiple times, police said.
The wounded officer, a 20-year veteran of the department, was in critical condition but was expected to survive, authorities said.
Edwards said police "stopped a tragedy that could have been a lot worse."
Four bodies were found inside the temple and three outside, including that of the gunman, officials said.
Of the possible motive for the shooting, Ri told the Journal Sentinel, "We have no idea."
The suspect was "not an insider," he said.
Dhaliwal told the newspaper the temple had not been the subject of any recent threats or graffiti.
"This is insanity," he said.
In the United States, where there are an estimated 314,000 Sikhs, Sikhs are sometimes mistaken for Muslims, especially since the Sept. 11, 2001, terrorist attacks.
Sikhs believe in the equality of humankind, universal human brotherhood and one supreme God.