Robert Bowers is suspected of storming a Pittsburgh synagogue and opening fire during a baby naming ceremony in an attack which killed 11 people.
Bowers, 46, was armed with what appeared to be an assault rifle and three handguns and is believed to have acted alone, according to authorities.
He was not known to law enforcement officials before Saturday's attack, said the FBI.
The bureau said Bowers murdered the worshippers and was then leaving the building when he came into contact with a uniformed police officer.
The pair exchanged gunfire and Bowers then went back inside before a SWAT team arrived, according to the FBI.
After a shootout, he surrendered.
He was taken to hospital with multiple gunshot wounds, which were not believed to be self-inflicted.
He is described as being in a fair condition.
Bowers now faces 29 charges, including using a firearm to commit murder.
The suspect, who had no apparent criminal record, appears to have expressed strong antisemitic views on a social media site called Gab.
Just hours before the attack, a social media post apparently by Bowers said a Jewish refugee organisation called the Hebrew Immigrant Aid Society (HIAS) "likes to bring invaders in that kill our people".
"I can't sit by and watch my people get slaughtered. Screw your optics, I'm going in."
According to the AP news agency, the cover photo for the account featured a neo-Nazi symbol.
And recent posts on the account reportedly included a photo of a fiery oven like those used in Nazi concentration camps used to cremate Jewish people during World War Two.
Other posts are believed to have referenced false conspiracy theories suggesting the Holocaust in which an estimated six million Jews perished was a hoax.
In another post, President Donald Trump was criticised for allowing Jewish people to enter the US.
Gab is a Philadelphia-based social networking service created as an alternative to Twitter.
In a statement, Gab.com confirmed the profile belonged to Bowers and also said it had suspended the account and contacted the FBI about it.
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HIAS is a non-profit group that helps refugees around the world find safety and freedom. The organisation said it is guided by Jewish values and history.
President and chief executive Mark Hetfield said he was not aware of the alleged gunman's "obsession with HIAS until this morning".