Now a Pentecostal pastor in Western Australia, Court caused controversy in May 2017 when she said she would boycott Qantas Airlines
for its decision to support same-sex marriage.
After being criticized by former and active tennis players for that stance, Court hit back by saying tennis was "full of lesbians
Speaking at a media conference ahead of the Australian Open, which gets underway Monday, King said that she "certainly didn't think they should have her (Court's) name anymore" on the 7,500-seat arena.
"I was fine until she said lately so many derogatory things about my community. I'm a gay woman ... that really went deep in my heart and soul," King added.
"If you were talking about indigenous people, Jews or any other people, I can't imagine the public would want somebody (with those views) to have their name on something. Maybe because of our community, the LGBTIQ community, people might feel differently," she said.
King, who herself won 12 major titles and faced Court in the 1969 Australian Open final, said she had initially been supportive of the decision to name the stadium in Court's honor.
But that view has changed in light of Court's comments. King also added that she would refuse to appear in the arena were she still playing today but would not encourage others to do so.
She instead encouraged others to "look into their heart" before making a decision.
Court has attended the Australian Open as a guest in the past but revealed to
Melbourne's Herald Sun newspaper earlier this month she would not be attending this year.
She also told the newspaper she did not hate gay people and was angered that she was made out to be homophobic.
Court is among the most successful players of all time having notched 24 major titles in a career that spanned the 1960s and 70s.
But she has also spoken out against Australia's recent decision to legalize same-sex marriage.
Speaking to the New York Times Thursday, former women's tennis legend Martina Navratilova said
she would refuse to play in the arena if she were still playing today.
"You do not name a building after her. Would you be naming a new building after her now? No, there's no chance," Navratilova said.
Navratilova, who is also gay, has previously called Court a "racist and a homophobe
CNN attempted to reach Court through her Victory International Centre church by phone and email but did not receive a response before publication.
Australian Open organizers have made it clear they do not agree with Court's views but have not pushed for a name change thus far.
The UK's Press Association reports that the issue is complicated by the fact that the Melbourne Park venue is managed and operated by a government trust.