Court, 74, has said she would not fly on Qantas "where possible" in protest at its support of same-sex marriage.
She then told a Christian radio station "tennis is full of lesbians".
In an open letter, Navratilova said: "We should not be celebrating this kind of behaviour."
The 60-year-old addressed her letter to the Margaret Court Arena, one of the main show courts at the Australian Open.
She said: "It is now clear exactly who Court is: an amazing tennis player, and a racist and a homophobe.
"Her vitriol is not just an opinion. She is actively trying to keep LGBT people from getting equal rights (note to Court: we are human beings, too)."
In 1990, Court said Navratilova was a poor role model for young tennis players because of her homosexuality.
Navratilova said she had forgiven Court for those comments, but had only just been made aware of remarks the Australian made about South Africa's apartheid regime.
In 1970, Court said: "South Africa has the racial situation rather better organised than anyone else, certainly much better than the United States."
Court won 24 Grand Slam singles titles, 11 of them in the Open era, which began in 1968 and allowed professionals to compete alongside amateurs.
Navratilova described Court's actions as "bullying" and said sporting venues are named after athletes for "who they are as human beings" and "not just for what this person did on the field".
"The platform people like Margaret Court use needs to be made smaller, not bigger," she said.
Navratilova believes the Margaret Court Arena should be renamed after Evonne Goolagong, a 14-time Grand Slam winner of Australian Aboriginal descent.
"I think the Evonne Goolagong Arena has a great ring to it," she added. "Now there is a person we can all celebrate. On every level."
BBC Sport tennis correspondent Russell Fuller:
Freedom of speech is one thing, but Margaret Court has caused widespread offence within tennis with these most recent remarks.
Tennis Australia has so far tried to separate Court's views from her achievements as a player to argue the name of the arena does not need to change. But they will find it very hard to withstand such pressure from figures like Martina Navratilova. And the current generation of players have much influence, too.
Andy Murray, who is a member of the ATP Player Council, says it would be difficult for players to boycott a particular court during a Grand Slam. But he points out they could collectively agree a position before the tournament, which would make life very difficult indeed for Tennis Australia.

Harry Gould, from Newcastle-under-Lyme, will be presented with the Legion d'Honneur by the Lord Lieutenant of Staffordshire.
Mr Gould served with the Royal Army Service Corps and was involved in the transportation of supplies to troops.
The charity that helped him apply said Mr Gould had been unaware he was eligible until told over a cup of tea.
Latest stories from Staffordshire
Mr Gould said he had been 18 years old when he landed at Arromanches, in Normandy.
"We had to come off the landing crafts. There were thousands of [men] just dead on the ground.
"The cargo we were carrying was mainly fresh water, ammunition and food.
"The water was important because the water in in Normandy had been polluted by the Germans."
Later, as Mr Gould was travelling in convoy, the two wagons in front of him exploded after being hit by aircraft fire.
"It made you feel terrible, but you had to keep going," he said.
He said he had been overwhelmed when learning he was eligible to receive the medal.
"It's like having an OBE in this country," he said.
Geoff Harriman, chairman of Tri Services and Veterans Support Centre, helped Mr Gould apply for the honour.
"Through conversations over a cup of tea, we found out Harry took part in the D-Day landings," he said.
"We knew the French government were awarding medals, so we applied on his behalf.
"It's France's highest award to non-French nationals. It's a way of saying thank you for assisting in the liberation of France."
Eighteen-time Grand Slam champion Martina Navratilova has renewed her call for the Margaret Court Arena to be renamed - describing the Australian as a "racist and a homophobe".
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A 91-year-old former soldier has been awarded one of France's highest honours for his role in the D-Day landings.