A day after the attack that killed an American contractor and wounded five US troops, which Washington blamed on a drone of Iranian origin, sources said there had been a new missile attack on another US base in northeast Syria.
White House national security spokesman John Kirby said it had been ineffective and there were no new US casualties.
Suspected US rocket fire later on Friday targeted new areas in eastern Syria, according to two local sources, with no casualties reported.
Pro-Iranian forces in Syria said in an online statement they had a "long arm" to respond to further US strikes on their positions.
The violence could further aggravate already strained relations between Washington and Tehran, as attempts to revive a 2015 nuclear deal between Iran and major powers stalled, and Iranian drones being used by Russia against Ukraine.
Although US forces stationed in Syria have been attacked with drones before, deaths are rare.
"Make no mistake: the United States does not ... seek conflict with Iran, but be prepared for us to act forcefully to protect our people," Biden told reporters during a visit to Canada.
Asked whether there should be a higher cost for Iran, Biden replied: "We're not going to stop."
The Pentagon had said US F-15 jets on Thursday attacked two facilities used by groups affiliated with Iran's Islamic Revolutionary Guards Corps.
The Syrian Observatory for Human Rights said the US strikes had killed eight pro-Iranian fighters.
Iran's state Press TV said no Iranians had been killed and quoted local sources as saying the target was not an Iran-aligned military post, but that a rural development centre and a grain centre near a military airport had been hit.
The US strikes were a response to a drone attack on Thursday on a base near Hasakah in northeast Syria operated by a US-led coalition battling the remnants of Islamic State.
Three service members and a contractor required medical evacuation to Iraq, while two wounded American soldiers were treated at the base.
The Pentagon said the injured personnel were in stable condition.
US officials said it appeared the defensive system on the base had failed.
A US base at the Al-Omar oilfield in Syria was attacked on Friday morning, according to the Lebanese pro-Iranian TV channel Al Mayadeen and a security source.
It is not uncommon for Iranian-backed groups to fire missiles at US bases in Syria after they are hit with air strikes.
US forces first deployed into Syria during the Obama administration's campaign against Islamic State, partnering with a Kurdish-led group called the Syrian Democratic Forces.
About 900 US troops are in Syria, most of them in the east.
