"Maybe it's time to change of the Margaret Court Arena then," wrote Navratilova, who won 18 Grand Slam singles titles, before adding in a later tweet: "Margaret - you have gone too far. Shame on you ... #wrongsideofhistory".
Navratilova and fellow tennis great Billie Jean King, who are both gay, have previously criticised Court for her views on homosexuality.
Melbourne's Margaret Court Arena, part of the Australian Open venue, was originally called Show Court One, when it opened in 1988, before it was renamed in 2003, in tribute to the multiple Grand Slam winner. It also hosts other sporting events and concerts.
And the official Twitter account for the venue even came out in opposition to Court's comments on Thursday night.
American singer Ryan Adams, who is playing at the arena on Friday, tweeted: "Hey Margaret Court, a PAGAN who STRONGLY SUPPORTS marriage equality (get a prenup tho) will play the arena named for you tomorrow".
The nation's other major domestic airline, Virgin, and founder Richard Branson also support gay marriage.
Qantas chief executive Alan Joyce recently had a pie thrown in his face in front of hundreds of people at a business breakfast in Perth by Tony Overheu, 67.
Overhu said it was in protest at what he said was "corporate bullying", including Mr Joyce and 20 chief executives of major companies signing a joint letter in support of marriage equality.
He was charged by police with assault and banned by Qantas.
Court, a 74-year-old senior pastor at Perth's Victory Life Church, and a critic of homosexuality and same-sex marriage for decades, said in the letter that she had proudly promoted Qantas during her days "of never losing a tennis match while playing for my country".
"I love all people and will be pleased to talk to your board at any time," she wrote.
"But it won't be in the Qantas lounge."
Court is the most decorated player in Grand Slam tennis history, winning a record 24 major titles from 1960 to 1973.
