Palestinians young and old held sit-ins and marches on Tuesday in the Israeli-occupied West Bank, the territory where the new law is most sweeping.
The Fatah political party has called a general strike in the northern part of the West Bank for Wednesday.
The legislation orders West Bank military courts - which try only Palestinians - to make the death penalty the default sentence for those convicted, except in special circumstances.
"Time is running out and silence is deadly," read the signs carried by protesters in the central West Bank city of Nablus.
"Stop the law to execute prisoners before it's too late."
The bill passed its final vote in the Israeli parliament late on Monday to cheers and applause.
Itamar Ben-Gvir, the Israeli firebrand minister of national security who spearheaded the push for the legislation, described the law as long overdue and a sign of strength and national pride.
The law is set to take effect in 30 days but its implementation could be delayed by pending court proceedings at Israel's highest tribunal.
The measure is not retroactive and will not apply to current prisoners.
Palestinian officials released statements saying the death penalty measure violated international law and asking other countries to intervene.
Amnesty International has said the use of the death penalty under the new measure could violate the right to life and the prohibition of torture and other cruel, inhuman or degrading treatment or punishment, as enshrined in international law.
The Palestinian foreign ministry called for sanctions on Israel's parliament and its suspension from international bodies.
"The law represents a critical turning point in the formalisation of extrajudicial killings under a legal guise," the statement said, adding that the ministry "stresses that this law, in its essence, constitutes an institutionalised policy of field executions based on discriminatory and racist standards".
The bill's passage was the culmination of a years-long push by Israel's far right to escalate punishment against Palestinians convicted of attacking Israelis.
A coalition of Israeli rights groups and opposition MPs announced they were launching a petition to Israel's Supreme Court to declare the law null and void.
In Gaza, dozens joined a demonstration in front of the headquarters of the Red Cross where women held up large framed photographs of well-known Palestinian prisoners such as Marwan Barghouti.
The law extends also to Israeli courts, giving them the option of imposing the death penalty on Israeli citizens convicted of nationalistic murder - language that legal experts say effectively confines those who can be sentenced to death to Palestinian citizens of Israel and excludes Jewish citizens.
