More than 50 Palestinians were killed in clashes with Israeli forces on Monday. It was the highest Palestinian single-day death toll since a series of protests dubbed the "Great March of Return" began at the border with Israel on March 30 and since a 2014 Gaza war.
Meanwhile, Israeli leaders and a U.S. delegation including Treasury Secretary Steven Mnuchin and President Donald Trump's daughter and son-in-law, Ivanka Trump and Jared Kushner, opened the embassy in Jerusalem, relocated from Tel Aviv.
Prince Turki al-Faisal of Saudi Arabia told CNBC earlier Monday that moving the embassy will make the Middle East a more dangerous place.
"It's not a step that will bring peace to Palestine or the Middle East," he said.
The United Nations says the status of Jerusalem — captured by Israel in the 1967 Middle East war — can only be resolved by negotiations. Palestinians want the city as their own capital.
Rubin said the embassy move provides a reason to protest, but believes there is much more to it than that.
For one, the leader of the Palestinian Authority, Mahmoud Abbas, is in his 80s and hasn't appointed a successor, Rubin said.
"The Palestinian clashes, especially from the Hamas-run Gaza Strip, have a lot to do with shaping the future of Palestinian politics and not just about the spark of the embassy move," he told "Power Lunch."
In fact, there are a lot of moving parts in the Middle East right now, along with general change underway, said Rubin.
"Every U.S. administration is often defined by the problem that no one saw coming," he said, like Bosnia for President Bill Clinton, Kuwait for President George H.W. Bush and Syria for President Barack Obama.
"Despite all the chaos right now, we still may get a problem which is coming in seemingly from left field."
— CNBC's Nyshka Chandran and Reuters contributed to this report.