Warren said that disagreeing with gay-rights activists on same-sex marriage does not qualify as hate speech and doesn't mean he is anti-gay. He said Mr. Obama chose him to give the invocation at the swearing-in to show that people with different views don't have to demonize each other.
"We're both willing to be criticized in order to try to bring America into a new day of civil discourse and to create a new model that says you don't have to agree only with your side on everything," Warren said in the video posted Monday night by Saddleback Community Church.
Gay-rights advocates were enraged that Mr. Obama had given the evangelical clergyman a prominent role at the Jan. 20 inauguration. Mr. Obama said he wanted the event to reflect diverse views and insisted he remains a "fierce advocate" of equal rights for gays.
Warren had backed Proposition 8, which banned gay marriage in his home state of California, where he founded Saddleback. He had recently said that he opposed any redefinition of marriage, including a brother marrying a sister, or an adult marrying a child.
In his video, he insisted he wasn't equating gay marriage with incest or child molestation.
"I have in no way ever taught that homosexuality is the same thing as a forced relationship between an adult and a child, or between siblings," Warren said. "I was trying to point out I'm not opposed to gays having their partnership. I'm opposed to gays using the term marriage for their relationship."
Perhaps hoping to improve his image, Warren paid an unannounced visit to a West Hollywood store that supports gay causes on Monday.
Erol Sarabi, the openly gay manager of the thrift store "Out Of The Closet," and Warren had their picture taken and chatted, reports CBS News correspondent Bill Whitaker.
Sarabi said Warren "kind of just wanted to let me know that his church had given a lot of money to people with AIDS and not to believe everything that I had read."
Howard Bragman, a gay activist and celebrity publicist, called Warren's West Hollywood shopping trip a publicity stunt.
"Am I suspicious?" Bragman said. "He just happens to show up at a thrift store in West Hollywood that benefits the HIVE and gay and lesbian community?"
On Tuesday, Warren's church replaced a brief article on the Bible and homosexuality with an audio message on Saddlebackfamily.com to better explain the church's view that Scripture prohibits sex outside of marriage between a man and a woman, according to Larry Ross, a Warren spokesman.
Anyone can attend Saddleback worship services. But the church article had said that gays "unwilling to repent of their homosexual lifestyle would not be accepted" as members.
Saddleback members must sign a broadly worded covenant in which they agree to follow Bible teachings. While gay relationships aren't mentioned in the pledge, it is meant to cover the spectrum of conservative Christian belief.