"The speaker, of all people, should certainly know what it's like to go to work every day afraid of being fired," Griffin said, a reference to the unsuccessful, tea party-backed challenge to Boehner earlier this year.
Twenty-two states and the District of Columbia have approved laws banning workplace discrimination on the basis of sexual orientation, and 17 of those also prohibit employers from discriminating based on gender identity.
About 88 percent of Fortune 500 companies have adopted nondiscrimination policies that include sexual orientation, according to the Human Rights Campaign. About 57 percent of those companies include gender identity.
The U.S. Chamber of Commerce remains neutral on the bill, a spokeswoman said Monday.
Drew Hammill, a spokesman for House Democratic Leader Nancy Pelosi, said it was disappointing that Boehner may not bring the measure to a vote. "When the Senate passes this legislation, all options will be on the table in order to advance this critical legislation in the House," Hammill said.
At the White House, spokesman Jay Carney sidestepped questions about whether Obama would consider issuing an executive order on workplace discrimination if Congress refused to act. Gay rights groups have criticized Obama for refusing to take that step, which would affect employees who work for federal contractors.
"We're focused on getting ENDA through Congress," Carney said, using the acronym for the workplace discrimination bill.
Associated Press writers Josh Lederman and Alan Fram contributed to this report.
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