An American soldier was killed by a rocket strike near Basra today, in the first US fatality since the last combat troops left Iraq.

The announcement came amid growing concern that the withdrawal of combat forces will allow security in Iraq to further deteriorate. The past three months have seen a spate of bombings and shootings in the centre and north of the country .

Details of the incident were not released, but Basra airport base, which is still home to about 4,000 US forces, had experienced increased numbers of rocket attacks in recent weeks as the deadline drew near for the withdrawal of combat troops. Two soldiers suffered minor wounds in a rocket strike early last week, and rockets have hit the Green Zone in Baghdad almost daily for the past month.

Around 50,000 US troops will stay in the country until next year, focusing mainly on training Iraqi security forces, but the top US military commander in Iraq said today they could return to combat operations if needed.

General Raymond Odierno told CNN the remaining troops could move back to combat if there was "a complete failure of the security forces", or if political divisions split the Iraqi security forces. "But we don't see that happening," Odierno said.

The last US combat brigade in Iraq crossed the border into Kuwait on Thursday, fulfilling President Obama's pledge to end combat operations by the end of this month.

The troops' departure has left many Iraqis apprehensive, with some predicting a rapid regression towards sectarian fighting.

Iyad Othman, a Fallujah policeman, said: "Now that they are gone for good, the situation will not continue to improve. They are surrendering the country to the Iranians."

In Baghdad the mood remains fearful. "It scares me to think that the Americans are leaving," said Umm Ali, 33, in the streets of Arasat, a well-to-do suburb. "If 144,000 soldiers could not control the situation here, how will things improve when there are only 50,000 here and they all stay in their camps?

"Obama did not think about Iraq's interests when he pressed ahead with this withdrawal," she said. "It is a very bad decision, and so is the timing. He should wait for a new government to be formed, one that is strong and can establish – then respect – the rule of law."

Ali Fidel, 40, a public servant in the agriculture ministry, was dismissive of US claims that Iraqi forces had been readied to fill the void left by departing US troops. He said: "If we take a look at the last two months in Iraq, security has now deteriorated to what it was in 2008. I think it will collapse after 31 August, and the country will then be in chaos.

"The Iraqi forces are not ready to work. They don't have the equipment to protect themselves, so how will they protect the civilians?"

Muner Salam, 50, a doctor from Mansour in west Baghdad, said: "Since 2003, Iraq has been destroyed day by day. Political parties and officials are too busy feuding over how to improve their own positions. The Americans cannot control it and now they decide to leave it to ruin.

"There is no government, no infrastructure, and daily explosions targeting doctors and judges. I am pessimistic. I think the country will face enormous challenges, and will not gain victories quickly. It will take a long time to control security."

General Odierno said today it could take years to determine if the US-led invasion was a success. "A strong, democratic Iraq will bring stability to the Middle East, and if we see an Iraq that's moving toward that, two, three, five years from now I think we can call our operations a success," he said.

• This article was amended on 24 August 2010. The original referred to 5,000 troops staying on in Iraq until next year. This has been corrected, and a correction published in the Corrections and clarifications column also on 24 August 2010.