Rivlin, whose post is mainly ceremonial, made the comments as unrest has spread in recent days, particularly in east Jerusalem and the West Bank.
Clashes have also rocked the Al-Aqsa mosque compound in Jerusalem, which is holy to both Muslims and Jews, who refer to it as the Temple Mount.
Israeli police have raided the site and fired tear gas and stun grenades at masked youths throwing stones and firebombs while barricading themselves inside the mosque itself.
"Those who wish to turn the tragedy between us, Palestinians and Israel... Into a religious war have blood on their hands," Rivlin told journalists.
He criticised a statement from Palestinian president Mahmud Abbas, who said last month that Palestinians will not allow Al-Aqsa to be spoiled by "filthy feet".
Palestinian officials later said he was referring to security forces who entered the mosque to shut the doors on rioters.
Rivlin also made reference to the radical northern branch of the Islamic Movement in Israel and others accused of orchestrating some of the protests at the site.
The Al-Aqsa compound is the third-holiest site in Islam and the most sacred in Judaism. It is located in east Jerusalem, annexed by Israel in 1967 in a move never recognised by the international community.
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