For the past week, Disney employees, contractors and fans, many of whom live in Florida and work at Disney World, have been pressuring the company on social media and in online petitions to take a public stand against the Parental Rights in Education bill. The measure, called by opponents the “Don’t Say Gay” bill, passed the Florida Senate on Tuesday. When Disney executives said they wouldn’t declare a stance, saying their inclusion efforts were best focused on representation in Disney programming, the outcry only grew louder.

On Wednesday, Mr. Chapek told shareholders: “We were opposed to the bill from the outset, but we chose not to take a position on it…because we felt we could be more effective behind the scenes.”

The shareholder meeting was Mr. Chapek’s first without his predecessor, longtime CEO

Robert Iger,

at the company. Mr. Chapek’s address to the gathering in Burbank, Calif., included details on his metaverse plans and the viral success of the animated film “Encanto.”

Yet, Mr. Chapek spent much of the meeting fielding questions about the company’s response to the Florida bill. The discussion highlights the complexities for executives leading a company when employees and fans all have personal social-media platforms to speak from and there is an expectation that executives will weigh in on charged issues.

Few political issues have engulfed a consumer-facing company with such speed, especially for Disney, which has long touted itself as an inclusive employer committed to putting LGBTQ stories on screen. The Florida bill prohibits discussion of sexual orientation and gender identity in certain grade levels at schools and allows parents to put limits on services provided at schools.

LGBTQ activists have said the bill will cause harm to children struggling with their sexual and gender identity by cutting off support from teachers and encouraging teachers to out gay students to their parents.

The bill’s sponsors have defended it as a way of letting families deal with children’s issues rather than teachers.

Mr. Chapek said Mr. DeSantis had agreed to meet with him and LGBTQ members of Disney’s senior leadership to discuss the company’s concerns about the bill. Disney has also signed onto a letter from the Human Rights Campaign disavowing the bill and said it would donate $5 million to LGBTQ causes, he said.

A spokeswoman for Mr. DeSantis confirmed the phone call, saying “this is the first time we have heard from Disney” regarding the legislation and that no in-person meeting had been scheduled yet.

“The governor’s position has not changed,” the spokeswoman said. “The same Florida parents who take their families to Disney also support parental rights in education, because they do not want their young children exposed to inappropriate content about sex and gender theory at school,” she added.

Later in the annual meeting, Mr. Chapek faced questions from shareholders about his response and his talk with Mr. DeSantis. He said he had an “extraordinary conversation” with the governor about how the legislation was drafted. Mr. Chapek said Mr. DeSantis suggested Disney executives look at the legislation as written and then discuss “which aspects were most concerning.”

On Monday, in response to the pressure, Mr. Chapek had sent an email to all Disney employees saying the company’s leadership team felt that it would be counterproductive for the company to take a public stance.

“As we have seen time and again, corporate statements do very little to change outcomes or minds,” Mr. Chapek said in Monday’s statement.

That response amplified critics who wanted to hear more from the large Florida employer. On Wednesday, Mr. Chapek told investors the company had been lobbying against the legislation.

“Despite weeks of efforts we were ultimately unsuccessful,” he said.

Inside Disney, senior executives are trying to calm employees upset about the law and Mr. Chapek’s reaction to it.

Some Disney employees said they noticed how Mr. Chapek’s desire to keep Disney out of political fights seemingly set him apart from Mr. Iger, who as CEO on occasion would speak out on controversial issues.

In 2019, Mr. Iger suggested Disney would stop production in Georgia in response to a restrictive antiabortion bill proposed in the state. Though he has left Disney, Mr. Iger also weighed in on the new Florida law, tweeting that “If passed, this bill will put vulnerable, young LGBTQ people in jeopardy.”

Ayo Davis, who as president of Disney Branded Television oversees content creation for Disney Channel, Disney+ and other outlets, sent a memo to her staff on Tuesday offering support.

“Know that we stand with the LGBTQ+ community and are firmly opposed to discrimination. You’re not alone,” Ms. Davis wrote.

—Erich Schwartzel contributed to this article.

Write to Robbie Whelan at robbie.whelan@wsj.com and Joe Flint at joe.flint@wsj.com