Mr Streeting, who was appointed to the frontbench role in November, said it is “perfectly legitimate” to disagree with the Edinburgh-based Harry Potter author, who has become a controversial figure over her comments on gender identity.
But he called for “a lot more empathy and understanding” when standing up for LGBT rights, warning “don’t try and shut down other people’s voices”.
Speaking to BBC Radio 4’s Political Thinking with Nick Robinson, Mr Streeting said: “I see myself as part of an LGBT community.
“I will always stand up for trans rights and for a community that doesn’t have representation in Parliament, is at the worst end of all the statistics on things like mental ill health, physical abuse, sexual abuse.
“I will always stand by them and speak up for them.
“But what I would also say to my own community is if we’re trying to win round hearts and minds and persuade people and to negotiate the future of LGBT equality, don’t try and shut down other people’s voices.”
In 2020, Rowling wrote an essay explaining how she was partly motivated to speak about transgender issues because of her experience of domestic abuse and sexual assault.
Mr Streeting added: “It’s perfectly legitimate for people to disagree with JK Rowling and her position on this.
“But you’ve got to ask yourself if you’re in the business of winning hearts and minds, if someone like JK Rowling has written a deeply personal piece in which she outlines very personal, personal, painful experience of abuse, do you really think the way to win an argument for trans equality is to is to try and shut her down when she’s made an argument in that context?
“So I think there’s got to be a lot more empathy and understanding, and it does cut both ways.”
Rowling, in 2020, responded to an article headlined “Creating a more equal post-Covid-19 world for people who menstruate”, tweeting: “‘People who menstruate.’ I’m sure there used to be a word for those people. Someone help me out. Wumben? Wimpund? Woomud?”
She denies she is transphobic and has defended her comments by saying: “If sex isn’t real, there’s no same-sex attraction.”
Her critics have included Harry Potter stars Daniel Radcliffe, Emma Watson and Rupert Grint.
Earlier this week, a secondary school that named a house after Rowling has dropped the title in light of the Harry Potter author’s “comments and viewpoints surrounding trans people”.
The Boswells School in Chelmsford, Essex, has changed the name of its red house from Rowling to Holmes – honouring Olympic gold medallist Dame Kelly Holmes.
A school newsletter in July 2021 said its six houses are “represented by British citizens who have excelled”.
It added: “However, following numerous requests by students and staff we are reviewing the name of our red house ‘Rowling’ and in light of JK Rowling’s comments and viewpoints surrounding trans people.
“Her views on this issue do not align with our school policy and school beliefs – a place where people are free to be."