The son of the late US supreme court justice Ruth Bader Ginsburg called a decision to give Elon Musk and Rupert Murdoch an award named for his mother a “desecration” of her memory.

Discussing protests made to the Dwight D Opperman Foundation, which gives the Justice Ruth Bader Ginsburg Woman of Leadership award, James Ginsburg told CNN: “I don’t want to speak to what our other plans might be if the foundation doesn’t see the wisdom of desisting and ending this desecration of my mother’s memory. But I will say that we will continue to fight this.”

The second woman appointed to the US supreme court, Ruth Bader Ginsburg spent 27 years as a justice, becoming a hero to American liberals. She died aged 87 in September 2020 and was replaced by Amy Coney Barrett, the third conservative justice installed by Donald Trump.

Ginsburg helped establish the award colloquially known as the RBG, saying it would honour “women who have strived to make the world a better place for generations that follow their own, women who exemplify human qualities of empathy and humility, and who care about the dignity and wellbeing of all who dwell on planet Earth”.

Previous recipients have included Barbra Streisand and Queen Elizabeth II.

Last week, the Dwight D Opperman Foundation announced a five-strong list it said was chosen from “a slate of dozens of diverse nominees” but which included just one woman.

That was Martha Stewart, 82, the lifestyle entrepreneur (and member of the first RBG award committee) who in 2004 was convicted of fraud and jailed for five months.

The men were:

• Musk, 52, the billionaire owner of SpaceX, Tesla and X, the platform formerly known as Twitter, through which he has taken increasingly rightwing political stances;

• Murdoch, 93 and the rightwing media baron owner of Fox News;

• Michael Milken, 77, a financier jailed on securities charges, pardoned by Trump and now a philanthropist;

• And Sylvester Stallone, 77, the star of films including the Rocky saga and the violent Rambo franchise.

The list prompted protests including a widely publicised letter to the foundation from a former Ginsburg clerk. Jane C Ginsburg, the justice’s daughter and a Columbia University law professor, called it “an affront to the memory of our mother”.

James Ginsburg, the founder and president of Cedille Records, a classical music label, told CNN he did not have “a clue” how the list of honorees was decided.

He said: “The original purpose of the award was … to recognise an extraordinary woman who has exercised a positive and notable influence on society and served as an exemplary role model in both principles and practice.

“And whether you want to discuss the wisdom of opening up that to men or not is one thing, but I think it would be hard pressed to apply that description to people like Elon Musk and Rupert Murdoch. And that’s why the family is so upset … the whole family and her clerk family …

“I’ve been contacted by people I know and people even that I don’t know about this, saying how upset they are. My sister even got a threatening letter and one of the things we want to do here is set the record straight. The family had nothing to do with this. We were not consulted. We are vehemently against this appropriation of our mother’s name and this insult to her legacy.”

The Opperman Foundation has said it intends to honour both men and women because Ruth Bader Ginsburg “fought not only for women but for everyone”. The Guardian contacted the foundation for further comment.

James Ginsburg said his mother would be “appalled” by honours given in her name to “people who pretty much stand against all the things that she stood for in terms of trying to … make the world a better place for people striving for equality and for a more inclusive world where everybody is treated with respect.

“I think one of her law clerks made a great analogy … it’s a little bit like … if somebody gave money to a university to build a physics lab and they built a football stadium instead. It so violates the purpose of what was intended here. And this is not what my mother signed on to when the award was first created …

“We can discuss the wisdom of each [nominee], but the two that obviously stand out here are Elon Musk and Rupert Murdoch.”

The two men did not immediately comment.

James Ginsburg said: “When you think of trying to create a more just society, which of course was mom’s ultimate goal, those are probably about the last names that would come to mind.”

• This article was amended on 18 March 2024 to correct a misspelling of Ginsburg.