The arrest comes after a wave of bomb threats to American Jewish institutions since the start of the year spread concern and political backlash in the United States.
The suspect is a 19-year-old resident of southern Israel and a dual US-Israeli citizen, police said.
An Israeli court on Thursday remanded him in custody until March 30, and he could be seen lifting his shirt over his face to hide his identity as photographers sought to film him.
His identity is under a gag order while the investigation continues. His motive was not yet clear.
Israeli media quoted his lawyer, Galit Bash, as saying "there is a concern that his medical condition affects his cognitive function".
"We asked the court to order a medical examination," she said.
A police statement said "the investigation began in several countries at the same time, in which dozens of threatening calls were received at public places, events, synagogues and community buildings that caused panic and disrupted events and activities in various organisations".
The investigation was undertaken in cooperation with the FBI "as well as other police organisations from various countries", it added.
Israeli police spokesman Micky Rosenfeld said the suspect was believed to be behind a range of threats against Jewish community centres and other buildings linked to Jewish communities in the United States in recent months.
It was not clear precisely how many threats he was accused of making.
He is also suspected of being behind similar threats in New Zealand and Australia.
In addition, Rosenfeld said he is suspected of a bomb threat to Delta Airlines in February 2015 which led to an emergency landing.
Police said he used voice-disguising technology when making calls. Authorities seized computer equipment and other items.
'Large-scale' probe
US Attorney General Jeff Sessions said in a statement that "today's arrest in Israel is the culmination of a large-scale investigation spanning multiple continents for hate crimes against Jewish communities across our country".
"The Department of Justice is committed to protecting the civil rights of all Americans, and we will not tolerate the targeting of any community in this country on the basis of their religious beliefs."
Israeli Public Security Minister Gilad Erdan said in a statement that "we hope that this investigation will help shed light on some of the recent threats against Jewish institutions, which have caused great concern both among Jewish communities and the Israeli government."
More than 100 bomb threats to Jewish institutions in the United States have accompanied fears over whether hate crimes and anti-Semitic acts have been on the rise in the country.
Some have said that the rise of Donald Trump as US president has encouraged the extreme right and emboldened hate groups.
But the arrest of a Jewish teenager over at least a portion of the threats is likely to complicate the debate over the issue.
His arrest is not the first linked to the threats in the United States.
On March 3, the FBI arrested a former journalist, Juan Thompson, 31, suspected of making a handful of the bomb threats to Jewish community centres and institutions around the United States.
The Anti-Defamation League, which monitors anti-Semitism in the United States, welcomed the arrest but called on Jewish communities to remain vigilant.
"Even though it appears that the main culprit behind the majority of these attacks has allegedly been identified, anti-Semitism in the US remains a very serious concern," it said.
"No arrests have been made in three cemetery desecrations or a series of other anti-Semitic incidents involving swastika graffiti and hate fliers."
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