The Council of American Islamic Relations (CAIR) demanded the apology last week after KFI 640 host Bill Handel made fun of the death of at least 363 Muslims during a stampede at the annual Hajj pilgrimage in Saudi Arabia.
According to CAIR, Handel imitated the people screaming and then joked that the Muslims at the pilgrimage should use a helicopter to monitor pilgrimage traffic, as is done with Los Angeles' freeway traffic.
On his programme on Monday, Handel said he would apologise, but only if CAIR agreed to denounce "all bombing or attacks where intended victims are innocent citizens," to acknowledge Israeli sovereignty and to verify that the group had never had connections with any terrorist group or sponsor.
If those conditions were met, "I'll be more than happy to apologise for offending people, I'll apologise all day long," Handel said in a live 12-minute response to CAIR's complaint on Monday.
He said the station had received hundreds of letters from all over the world demanding an apology.
The organisation's spokeswoman Sabiha Khan said: "CAIR's record is clear. We've had numerous statements denouncing terrorism and all the things he was talking about."
She called Handel's conditions a "smoke screen" for his refusal to take responsibility for his remarks.