Witnesses said the man fired at least 12 bullets before he finished his ammunition and was overpowered by police and arrested.
The attack took place on Monday afternoon in a crowded part of central Amman close to a Roman amphitheatre - an area popular with tourists.
Nasser Joudeh, a government spokesman, said that two British people, a Dutch citizen, a New Zealander, a Dutchman and their Jordanian tour guide were wounded during the shooting.
Jordanian officials initially branded the attack a "terrorist act", however Joudeh was later quoted by Jordan's state-run Petra news agency as saying that the gunman had acted alone and had no ties to any "armed or suspicious group".
"The result of the primary investigation is that this was a lone act. He does not have any connections with terrorist organisations," he said.
The gunman has been identified as Nabil Ahmad, a Jordanian in his late thirties and a resident of the industrial town of Zarqa, on the eastern outskirts of the capital, Jordanian officials said.
The shooting is the first attack in the pro-Western kingdom since scores of people died in triple suicide bombings in luxury hotels claimed by al -Qaeda last November.
Muhammad Jawad Ali, an Iraqi man who witnessed the shooting, said: "I was walking when I saw someone pull out a pistol from his pocket and start shouting Allahu Akbar [God is Great] and fire repeatedly.
"Then I saw one tourist who appeared to be dead and three who were injured. They were in a group of seven. A woman told me they were tourists from New Zealand and England."
Israel's offensive in the Gaza Strip and war in Lebanon, which has killed some 1,400, has raised tensions in Jordan where anti-Israeli feelings are running high.
Many Jordanians are angry about what they see as Western indifference towards the plight of Palestinians.