The suspect accused of gunning down two Israeli embassy staffers outside a Jewish museum in Washington has been charged with murder, as international tensions over anti-Semitism erupted over the attack.

Elias Rodriguez, 31, shouted "Free Palestine" as he was taken away by police after the shooting late on Wednesday outside the Capital Jewish Museum.

"I did it for Palestine, I did it for Gaza," he told the officers.

The Chicago man made an initial court appearance on Thursday after being charged with two counts of first-degree murder and murder of foreign officials. If convicted, he could receive the death penalty.

Authorities were investigating the shooting "as an act of terrorism and as a hate crime", Jeanine Pirro, interim US attorney for the District of Columbia, said.

"I suspect as we go forward... that there will be more charges added," she said, noting that a preliminary hearing was set for June 18.

The shooting triggered international outrage and finger-pointing as Israel's foreign minister Gideon Saar blamed European criticism of his country's stepped-up Gaza offensive, claiming "a direct line connecting anti-Semitic and anti-Israeli incitement to this murder".

"This incitement is also done by leaders and officials of many countries and international organisations, especially from Europe," he said.

French foreign ministry spokesman Christophe Lemoine called the accusation "completely outrageous and completely unjustified".

Israeli Prime Minister Benjamin Netanyahu cited "the terrible price of anti-Semitism" and decried "wild incitement against the State of Israel".

Soon after the shooting, US President Donald Trump – who spoke with Netanyahu on Thursday – posted on social media that the attack was clearly anti-Semitic.

The killings took place outside the Capital Jewish Museum, located 1.6 kilometres from the White House, following a social event hosted by the American Jewish Committee for young Jewish professionals and the Washington diplomatic community.

Israel identified the victims as Yaron Lischinsky, an Israeli citizen, and Sarah Lynn Milgrim, a US employee of the embassy, and said they were a couple planning to marry.

The attack came days after the museum was awarded a grant to boost security, as anti-Semitism surges worldwide in the wake of Israel's devastating invasion of Gaza, prompted by the October 7, 2023, attack by Palestinian militant group Hamas.

Tensions have risen in the United States and many other countries over the ongoing Israeli assault on Gaza, with pro-Palestinian activists decrying what they say is the intolerable human cost of the military offensive.

Britain and France – who have stepped up their criticism of Israel's actions – were among those condemning the shooting, as well as Germany and the United Arab Emirates.

But Netanyahu on Thursday accused France, Britain and Canada of emboldening militants, saying "they want Israel to stand down and accept that Hamas's army of mass murderers will survive, rebuild and repeat the October 7 massacre".

Police said Rodriguez was seen pacing outside the museum before the shooting around 9 pm.

According to court documents, Rodriguez approached his victims, who were facing away from him, and fired 21 rounds.

He shot multiple times at the couple after they were already on the ground and fired at Milgram as she tried to crawl away.

Witnesses said security personnel at first mistook the gunman for a victim of the shooting and allowed him into the museum, where he was initially comforted by bystanders.

"They sat him down. 'Are you OK? Were you shot? What happened?' And he's like 'Somebody call the cops'," Yoni Kalin, who was in the museum, told US media.

Lischinsky was a researcher at the Israeli embassy, while Milgrim worked for its public diplomacy department, according to their LinkedIn profiles. (AFP)