US-based Human Rights Watch identified the rebel as Abu Sakkar, a well-known insurgent from the city of Homs, and said his actions were a war crime.
The main Syrian opposition coalition said he would be put on trial.
The video, which cannot be independently authenticated, 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 dog," 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 Free Syrian Army (FSA) Al-Farouq Brigades. He insults Alawites, the minority offshoot of Shia Islam to which Mr Assad belongs.
"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 Mr Bouckaert. "The danger is that extremists on both sides will feel the need to respond in kind."
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 - though a surgeon told the magazine that the organ appeared to be a lung, not the heart or liver, as reported elsewhere.
Syria's opposition coalition said the Al-Farouq Brigades were investigating the incident.
"The Syrian Coalition strongly condemns this act - if it is revealed to be true," it said in a statement. "The culprit will eventually be tried in court in front of an honest and fair judiciary."
It is one of the most gruesome videos to emerge in more than two years of carnage in Syria, says the BBC's Jim Muir in Beirut.
The UN says nearly 80,000 people have been killed since the uprising against President Assad began in March 2011, and millions have fled their homes.
By Jim MuirBBC News, Beirut
The results of many terrible atrocities in Syria have surfaced on internet video clips over the past two years and more. But this one was particularly chilling because it was filmed as it happened, showing an exultantly brazen and barbaric act of gratuitous bloodthirstiness.
Even if it was not intended as a provocation, it is likely to have that effect as the struggle between rebel and government forces for control of Quseir and nearby strategic villages intensifies.
The video performance - for that is what it was - had a strong element of sectarian incitement, as the leader identified as "Abu Sakkar" urged his followers to cut out and eat the hearts of Alawites (Bashar al-Assad's ruling minority).
It is likely to bolster the conviction among some that the regime is less bad than the possible alternatives, while weakening the argument of those who want to arm the rebels more seriously.
It is unlikely to have a direct impact on preparations for a peace conference sponsored by Washington and Moscow, which are already running into problems. But it may bolster the view of some US officials that it is better to push for a political settlement than to help the rebels win.
Meanwhile, Jordan has said it will host a meeting next week for the Friends of Syria - a group of Western and Arab nations that want Mr Assad to relinquish power - to help prepare for a peace conference on Syria in June.
US Secretary of State John Kerry and Russian President Vladimir Putin agreed last week to hold the conference, which they initially hoped to organise before the end of May.
The conference will try to persuade the Syrian government and opposition to accept a deal based on the core elements agreed by the UN-backed Action Group for Syria in June last year.
These included an immediate cessation of violence and the establishment of a transitional government that could include officials serving under President Assad and members of the opposition.
Syrian Information Minister Omran al-Zoubi said on Tuesday that Syria would need more information before deciding whether to attend such a conference, but would not be party to any meeting which harms "national sovereignty".
He said the future role of Mr Assad was an issue "only for the Syrian people and the ballot box".
Mr Kerry suggested the Syrian government would be at the table as it had already given a list of representatives to Russia.
"If he decides not to come to the table, it would be another one of President Assad's gross miscalculations," he said. "I don't believe that that is the case at this moment."
Mr Kerry said he had also spoken to Free Syrian Army leader General Salim Idriss, who had assured him he was committed to the negotiation process.
By Jonathan MarcusBBC diplomatic correspondent
Gruesome video pinpoints West's dilemma
Russian President Vladimir Putin, who has argued that Mr Assad could still have a post-war role, said it was "especially important to avoid any moves that could aggravate the situation".
"Only a quick cessation of hostilities and a political settlement can prevent a negative scenario," he said after talks with Israeli PM Benjamin Netanyahu.
Russia is concerned by Israeli air strikes on targets inside Syria, while Israel is unhappy at shipments of Russian weapons to Damascus.
The two leaders gave no indication of whether they had discussed Russian supplies of advanced weapons systems to Syria, including air defence missiles and artillery systems.
Last week, Russian Foreign Minister Sergei Lavrov said Moscow was not violating any international sanctions and would honour contracts already signed, but avoided confirming reports that it was preparing to sell Damascus S-300 air defence batteries.
A video which appears to show a Syrian rebel taking a bite from the heart of a dead soldier has been widely condemned.