The City Council speaker, Christine C. Quinn, has been a staunch supporter of New York City’s horse-drawn carriage industry, which has come under fierce criticism from animal rights advocates who believe it should be shut down.

But on Tuesday, Ms. Quinn issued a strong rebuke to the industry after the tennis player Martina Navratilova wrote her a letter calling on her to support a ban on the carriages not only to prevent animal cruelty, but also as a matter of protecting gay rights.

In a letter on Monday, Ms. Navratilova described an episode last month on the weekend of the city’s gay pride celebration, caught on video, in which a driver uttered anti-gay and racist slurs at a group of women who were handing out leaflets near a row of horse-drawn carriages.

Animal rights advocates have long singled out Ms. Quinn as they lobbied for a ban on horse-drawn carriages, with little success. But Ms. Navratilova faced no such trouble: On Tuesday, Ms. Quinn said the union representing carriage drivers had agreed to immediately require its members to undergo anti-discrimination training. Ms. Quinn said her office would work with the union to find an appropriate gay-rights organization to help put the training in place.

Ms. Navratilova, like Ms. Quinn, is gay. In her letter, which was first published on the Towleroad blog on Tuesday, Ms. Navratilova lamented the “aggressive homophobia exhibited by carriage drivers” and pointed out that a large number of animal rights activists are also gay.

“We share a heightened sense of responsibility in fighting the callous disregard that many have for animal suffering because we have experienced a similar disregard for the cruelty that we ourselves have faced,” she wrote, adding, “As evidenced by the hateful, bigoted carriage driver in the video, everything about this industry is stuck in the 19th century.”

Ms. Quinn, who is a likely, and leading, candidate to run for the Democratic nomination for mayor in 2013, quickly condemned the treatment of the leaf-letters in a sternly worded letter to the carriage drivers’ union, Teamsters Local 553.

“The behavior depicted in this video is reprehensible and unacceptable from anyone, and is especially unbefitting of an industry the City Council has made sure treats its animals humanely,” Ms. Quinn wrote. (Union officials did not return a phone message.)

In a separate letter to Ms. Navratilova, Ms. Quinn reiterated her support for the continued existence of horse-drawn carriages, saying she supported “allowing an industry that generates vital jobs and tourism for our city to continue.” But she thanked Ms. Navratilova for “being such a strong voice for animals and people everywhere” and told her that the union had agreed to put in place the anti-discrimination training.

“We must make sure that this doesn’t ever happen again,” she wrote.