Human Rights Watch identified the terrorist as Abu Sakkar, a well-known insurgent from the city of Homs, and said his actions were a war crime.
The video seems to show him cutting out the heart.
"I swear to God we will eat your hearts and your livers, you soldiers of Bashar," the man says, referring to President Bashar al-Assad as he stands over the soldier's corpse.
Human Rights Watch (HRW) says Abu Sakkar is the leader of a group called the Independent Omar al-Farouq Brigade, an offshoot of the so-called Free Syrian Army (FSA) Al-Farouq Brigades. He also insults Alawites.
"The desecration and mutilation of a killed person is definitely a war crime," Peter Bouckaert, emergencies director of Human Rights Watch, told the BBC. "This one particularly disturbing because of the sectarian nature of the language used by Abu Sakkar."
HRW said those committing war crimes on either side had to know that there was no impunity and that they would be brought to account.
The human rights group said Abu Sakkar had been filmed before, firing rockets into Shia areas of Lebanon and posing with the bodies of guerrillas from the Lebanese Hezbollah movement killed fighting alongside Syrian government forces.
"Abu Sakkar is a very significant commander - he's in charge of one of the most important battles happening in Syria right now," said Bouckaert.
The video was posted on Sunday, though reporters from Time Magazine said they had first viewed the footage in April.
Time said on Tuesday it had spoken to Abu Sakkar, who confirmed that he had bitten into one of the soldier's organs.
It is one of the most gruesome videos to emerge in more than two years of carnage in Syria.
Syria has been struggling with a Western-backed insurgency which began in March 2011 as a protesting movement but turned to be a cover for terrorist groups including al-Qaeda and al-Nusra Front.
The United States, Israel, UK and their allies have been supporting the armed insurgency in Syria, mulling ways to supply the militants with more arms to topple the legitimate government of Syrian President Bashar al-Assad.