(Updates with lowered alert level, lone wolf theory, Italy link)
By Philip Blenkinsop and Marine Strauss
BRUSSELS, Oct 17 (Reuters) - A Tunisian gunman suspected of killing two Swedish football fans in Brussels died on Tuesday after being shot by police in a cafe, hours after an attack which Sweden's prime minister said showed Europe must bolster security to protect itself.
The 45-year-old, who identified himself as a member of Islamic State and claimed responsibility in a video posted online, is also suspected of wounding another Swedish national in his attack in central Brussels on Monday evening.
"This is a time for more security, we can't be naive," Swedish Prime Minister Ulf Kristersson told a news conference in Stockholm, calling for stricter border controls in Europe. "These terrorists want to scare us into obedience and silence. That will not happen," he said.
The shooting came at a time of heightened security concerns across much of Europe linked to the Israel-Hamas conflict, but prosecutors said the gunman appeared motivated more by Koran burnings in Sweden.
In August, Sweden raised its terrorism alert to the second-highest level and warned of an increase in threats against Swedes at home and abroad after burnings outraged Muslims and triggered threats from jihadists.
Chief prosecutor Frederic Van Leeuw said early indications pointed toward the gunman operating alone, rather than being part of a broad network, although two people were brought in for questioning.
Police found an ArmaLite AR-15 semi-automatic rifle and a bag of clothing in the cafe where he was discovered on Tuesday morning. Two handguns and a knife were found very close to his apartment, Van Leeuw said.
Video footage of the attack showed a man in an orange jacket on a scooter at a street intersection with a rifle following people fleeing into a building and firing at them.
"I saw the assailant enter the building and shoot twice towards the man - the man fell to the ground. I saw him fall because I was just nearby," said a witness, who identified himself as Souleymane.
"I stayed there, I was frozen, I couldn't move. I'm still shivering because of what happened."
According to a media transcript of the video message recorded by the self-declared perpetrator, the attacker said he had killed Swedes to avenge Muslims.
The victims were a man aged around 70 from the Stockholm region, and a man aged about 60 who lived abroad, Sweden's foreign ministry said, adding that the injured man, also around 70, was still in hospital.