"I don't see why anyone has a problem with two people who love each other getting married," Murray told reporters after the Briton beat Andrey Kuznetsov in the first round of the French Open on Tuesday.
"If it's two men, two women, that's great. I don't see why it should matter. It's not anyone else's business. Everyone should have the same rights."
Retired 18-times grand slam champion Martina Navratilova reacted by suggesting Court's name should be taken off Melbourne Park's number two show court, while Stosur said that players might boycott the venue during the Australian Open.
"If something was to be done, I think it would be a lot more beneficial to do it before the tournament starts," Murray said.
"For players to be in a position where you're in a slam and kind of boycotting playing on the court, I think would potentially cause a lot of issues.
"So I think if something was going to be happening and the players come to an agreement, if they think the name should be changed or whatever, that should be decided before the event -- before the event starts."
Murray will face Martin Klizan of Slovakia in the second round on Thursday.
(Reporting by Aditi Prakash in Bengaluru; Editing by John O'Brien)
