Amir Fuchs, who heads the Defending Democratic Values programme at the Israel Democracy Institute, said: "There is no country in the world that has not specifically enumerated the right of equality in its Constitution - therefore, it is hard to understand why the authors of this Bill insist not to include this important value".
In a statement following the law's passage, Mr Odeh "today I will have to tell my children, along with all the children of Palestinian Arab towns in the country, that the state has declared that it does not want us here".
In 1947, after the end of World War II which saw the extermination of more than six million Jews, the United Nations adopted a plan to split Palestine into two states, one Jewish and one Arab.
The issue of Israel as a Jewish state has become increasingly important in recent years and a key dispute between Israel and the Palestinians.
But the bill states that Hebrew is the only official language of the state, demoting the status of Arabic. Critics contend the bill's expansive language would be devastating to Israel's image and undermine its positioning as being both Jewish and democratic.
"Long live the state of Israel", Prime Minister Benjamin Netanyahu said after the law's passage.
The question of Israel's status as a Jewish state is politically controversial and has always been debated.
The legislation passed after a clause was changed that would have allowed the state to "authorise a community composed of people having the same faith and nationality to maintain the exclusive character of that community".
"This is a creeping apartheid law that is just the beginning of a series of discriminatory laws against the Palestinian citizens of Israel", said Thabet Abu Rass, co-director of the Abraham Fund Initiatives, an Israeli NGO that works to promote equality.
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The Nation-State Basic Law, which has constitutional status, is anti-democratic: The law negates the main goal behind the introduction of a democratic constitution according to which residents living in a given territory are equal citizens and constitute the sovereign.
The "nation state" law, backed by the right-wing government, passed by a vote of 62-55 and two abstentions in the 120-member parliament after months of political argument.
'So today we have made a law in stone. "The Jewish Nation-State bill officially legalizes apartheid and legally defines Israel as an apartheid system", he said in a tweet.
Turkey and Israel, former allies, expelled each other's top diplomats in May during a row over clashes when dozens of Palestinians were killed by Israeli forces on the Gaza border.
Reacting to the law at a news briefing on Thursday, a spokeswoman for the EU's foreign affairs chief, Federica Mogherini, said, "We are concerned, we have expressed this concern and we will continue to engage with Israeli authorities in this context".
The nation-state law is perhaps the most contentious among them.
But opponents to the law say it marginalises the country's Arab minority.
The American Jewish Committee said it is "deeply disappointed" by the law, first because it "downgrades" Arabic and second because it promotes "Jewish settlement".
"Israel has a long and commendable record of respect for and protection of the rights of all its citizens including its ethnic and religious minorities, but provisions in the law could impact these protections", he added.