The Israeli military said the man had tried to seize a soldier's gun at the Allenby bridge, which spans the Jordan River, and that troops had then shot him.
The Palestinian Authority based in the West Bank city of Ramallah condemned what it called the "shooting at close range" and demanded an investigation.
A statement from Prime Minister Rami al Hamdallah's office identified the man as Raed Alaa el-Deen Za'eiter, 38, a West Bank-born judge working in Jordan.
An Israeli military official said the military were checking the reports about the man's identity. The crossing was temporarily closed.
The judge's father, Alaa Zueter, said he had gone to court to see his son today but didn't find him at work and only later heard on the media what had happened.
"I didn't know my son went to the West Bank today," said the man, crying over the telephone. He said he did not know what his son's business was in the Palestinian territory.
The border terminal is operated jointly by Israel, Jordan and the Palestinian Authority.
It is the main border crossing for Palestinians from the West Bank travelling to neighbouring Jordan and beyond, and it is a crossing point for goods between Israel, the West Bank and Jordan.