Kassig, who took the name Abdul-Rahman after converting to Islam, was captured last year and was threatened in an October 3 video showing the beheading of British aid worker Alan Henning. He was 26 years old.
"Abdul-Rahman was taken from us in an act of pure evil by a terrorist group that the world rightly associates with inhumanity," Obama said in a statement released by the White House aboard Air Force One as he returned home from an Asia tour.
The video also showed the gruesome simultaneous beheadings of at least 18 men described as Syrian military personnel, the latest in a series of mass executions and other atrocities carried out by IS in Syria and Iraq.
"Like (US journalists) Jim Foley and Steven Sotloff before him, his life and deeds stand in stark contrast to everything that ISIL represents," Obama added, using another acronym by which IS is known.
"ISIL's actions represent no faith, least of all the Muslim faith that Abdul-Rahman adopted as his own."
In the undated video, a man who appears to be the same British-accented jihadist who beheaded previous Western hostages stands above a severed head.
"Here we are burying the first American crusader in Dabiq, eagerly waiting for the remainder of your armies to arrive," the black-clad masked executioner says, referring to a northern Syrian town.
He also urges Obama to send more troops back to the region to confront IS, just as Washington prepares to double its military personnel in Iraq to up to 3,100 as part of the international campaign it heads against IS.
- 'Incredibly proud' -
Kassig, a former US soldier, had risked his life to provide medical treatment and aid to those suffering from Syria's civil war.
He founded a group through which he trained some 150 civilians to provide medical aid to people in Syria. His group also gave food, cooking supplies, clothing and medicine to the needy.
"We are incredibly proud of our son for living his life according to his humanitarian calling," his parents wrote, using the Twitter handle @kassigfamily.
US Secretary of State John Kerry said that American government officials worked alongside Kassig's family to try to secure his release.
"During his time in captivity, his family, and the entire government, including his home state Senator Joe Donnelly, worked to avoid this tragic outcome," the top US diplomat said.
Kassig was the fifth Western hostage killed by IS in recent months, after the two US reporters and two British aid workers were beheaded.
Sunday's video was substantially different from previous IS recordings. Kassig was not shown alive in the footage, and no direct threats were made against other Western hostages.
British Prime Minister David Cameron said he was "horrified" by the "cold-blooded murder," which French President Francois Hollande called a "crime against humanity."
Burhan Mousa Agha, a Syrian friend who worked with Kassig in Lebanon, described him as a funny, dedicated and brave man who only wanted to help.
"I want to apologise to his family. I'm sorry that their son died in my country, trying to help," Agha told AFP.
"They are animals, less than animals, they don't represent Islam. Peter wasn't fighting anyone, he was teaching people how to save lives.
"I want to send a message to everyone that Peter was a hero, a real hero," Agha added.
- Sequence of beheadings -
The Sunni Muslim extremist group has seized large parts of Iraq and Syria, declared a "caliphate" and imposed its brutal interpretation of Islamic law.
The latest video also showed a highly choreographed sequence of the beheadings of at least 18 prisoners said to be Syrian officers and pilots loyal to President Bashar al-Assad.
Jihadists are shown marching the prisoners past a box of long military knives, each taking one as he passed, then forcing them to kneel in a line and decapitating them.
"Today, we're slaughtering the soldiers of Bashar," the same black-clad militant said in the video.
"Tomorrow, we'll be slaughtering your (US) soldiers. And with Allah's permission, we will break this final and last crusade, and the Islamic State will... begin to slaughter your people on your streets."
The video came as IS suffered battleground losses in Iraq backed by US-led air strikes, with government forces Saturday breaking the jihadists' months-long siege of Iraq's largest oil refinery.
The Syrian Observatory for Human Rights monitoring group said IS also executed 13 men in Raqa in Syria, accusing them of belonging to pro-regime sleeper cells.