The group, which has captured territory in Syria and Iraq, released a video on Tuesday of Sotloff being beheaded. U.S. officials confirmed its authenticity on Wednesday. President Barack Obama vowed to "degrade and destroy" the group.
Barak Barfi, a friend of Sotloff who is serving as family spokesman, began a prepared statement from the family in English, remembering the slain journalist as a fan of American football who enjoyed junk food, the television series "South Park" and talking to his father about golf.
The 31-year-old Sotloff was "torn between two worlds," the statement said, but "the Arab world pulled him."
"He was no war junkie ... He merely wanted to give voice to those who had none," Barfi said outside the family's one-story home in a leafy Miami suburb.
Barfi ended the statement with off-the-cuff remarks in Arabic, saying "Steve died a martyr for the sake of God."
He then challenged Islamic State leader Abu Bakr al-Baghdadi to debate Islam, saying, "Woe to you. You said the month of Ramadan is the month of mercy. Where is your mercy"
"God does not love the aggressor," added Barfi, who is an Arabic scholar and research fellow at the New America Foundation think tank in Washington.
He went on, "I am ready to debate you with kind preachings. I have no sword in my hand and I am ready for your answer."
The other American hostage killed in recent weeks in retaliation for U.S. air strikes against Islamic State forces in Iraq was journalist James Foley, who was shown being beheaded in a video released on Aug. 19.
Sotloff was a freelance journalist who traveled the Middle East writing for the magazines Time and Foreign Policy, among others.
"Steve was no hero," the family said in its statement.
"Like all of us, he was a mere man who tried to find good concealed in a world of darkness. And if it did not exist, he tried to create it. He always sought to help those less privileged than himself, offering career services and precious contacts to newcomers in the region."
Sotloff was kidnapped in Syria in August 2013 after he drove across the border from Turkey.
He grew up in the Miami area and studied journalism at the University of Central Florida. A spokesman for Israel's foreign ministry said on social media website Twitter that Sotloff also was an Israeli citizen.
Sotloff "yearned for a tranquil life where he could enjoy Miami Dolphins games on Sunday," his family said.
"This week we mourn," it added. "But we will emerge from this ordeal ... We will not allow our enemies to hold us hostage with the sole weapon they possess - fear."