After he spoke, Israel's Shin Bet security service said troops shot and wounded a suspected militant who tried to evade arrest near the Palestinian city of Ramallah.
Israeli sources say two people have also sustained serious injuries in the shooting.
The Israeli military has also confirmed that another Palestinian was killed during an arrest attempt in the city of Nablus.
The Israel Defense Forces said in a statement that the gunman exited his vehicle and opened fire on soldiers and civilians standing at a bus stop at the Asaf Junction.
Local resident Huwayda Bushkar said she saw Israeli soldiers carrying a body as they left. "We believe he was taken to another area and extra-judicially executed". Barghouthi's father and brothers were reportedly detained. "But he was at home normally going to work normally, nothing was odd about him or his activities".
In two other incidents on Thursday, police said a Palestinian had stabbed and wounded two Israeli police troopers in East Jerusalem before being shot dead, while the army said a Palestinian driver had carried out a ramming attack in the West Bank targeting Israeli soldiers, injuring one of them.
"The security forces are in pursuit of the murderer".
The house where Naalwa was killed, located in the northern West Bank refugee camp of Askar, was peppered with bullet holes after the operation, an AFP journalist at the scene said.
The IDF said it is searching for the shooter in the nearby village of Silwad and has set up checkpoints.
"Hamas and the other resistance factions will continue to support activity to avenge their death", the statement added.
Two Border Police officers were stabbed and lightly wounded in a terror attack before dawn Thursday in the Old City of Jerusalem, before shooting dead their assailant, police and medics said.
Barghouti was identified by Palestinian media as a 29-year-old from the Palestinian village of Kobar and the son of Omar Barghouti*, a senior Hamas member and perennial security prisoner.
Many Palestinians consider violence against Israelis in the West Bank a justified response to the growth of settlements on land they see as theirs.
Israel accuses Naalweh of shooting to death two Israelis and wounding another at an attack in a West Bank industrial zone in October. One of those injured has now died. The IDF believes that the attacker then fled the scene by vehicle, driving somewhere near Ramallah.
The recent violence reflects current levels of tension.
Thursday's shooting occurred at a location about a 10-minute drive south from the place of Sunday's attack.
Around 600,000 Israelis now live in settlements there considered illegal by the global community.
Akram al-Waara contributed reporting from the West Bank.