A pool report from a reporter in the room said Obama "left the lectern and moved over to the protester." The pool report quoted Obama as saying: "Listen to me or you can take the mike, but I'm leaving. You all decide. You have one choice."
Obama's suggestion that she would leave was not included in the official White House transcript.
The audience responded by asking her to remain, according to the pool report, which quoted a woman nearby telling Sturtz, "You need to go."
Sturtz was escorted out of the room. She said in an interview later that she was stunned by Obama's response.
"She came right down in my face," Sturtz said. "I was taken aback."
Sturtz said she told Obama she was happy to take the microphone to plead her case, which, Sturtz said, appeared to fluster the first lady.
"I said I want your husband to sign the executive order," Sturtz said. "Her husband could sign this order tonight and protect 22 percent of the workforce in this country."
Sturtz said she paid $500 to attend the fund-raiser, part of a protest organized by the gay-rights group GetEqual, which gained notice in the president's first term for hectoring him during speeches and demanding more action on lesbian, gay, bisexual, and transgender issues.
Sturtz, who gave $5,000 to the Democratic Party and the Obama campaign in 2008, according to the Center for Responsive Politics, said she was devoting herself to full-time activism now pressing the White House on the employment-discrimination issue.
The proposed executive order that prompted Sturtz's outburst would prohibit federal contractors from discriminating on the basis of sexual orientation or gender identity.
LGBT activists, many of whom hailed the president for his vocal support for same-sex marriage rights in the months leading up to his reelection in 2012, have been increasingly dismayed that the White House has not yet acted on the proposed order.