Gharsalli said officials were still verifying whether the attacker had been trained in camps in neighboring Libya.
"We will find all those involved, whether it was just logistical support or not," Gharsalli said, flanked by ministers from Britain, France and Germany.
British Prime Minister David Cameron on Monday announced a national moment of silence to be held in honor of the British tourists slain.
Speaking in the House of Commons, Cameron announced that the commemoration would be held at midday Friday. He also said a memorial is planned for the victims in the future.
Eighteen Britons are confirmed dead, but officials have warned that the U.K. toll could rise as high as 30, making it the worst attack on British citizens since the London transport attacks that killed 52 commuters in July 2005.
Cameron said the government wasn't advising Britons not to travel to Sousse and other Tunisian holiday destinations, though that will be kept under review.
He said the global nature of the struggle against radicalism meant "nowhere is without risk from Islamist extremist terrorists."
Thousands of tourists have left Tunisia since Friday's attack, which has shocked a country that relies heavily on tourism for jobs and foreign currency revenues.