The Conservative leadership candidate said the UK does not “need to get caught up” in a culture war on the issue.
The Transport Secretary’s comments came as Tory leadership contenders set out their stall for succeeding Boris Johnson as Prime Minister.
Fellow leadership hopeful Tom Tugendhat called for a “clean start” in the trans rights debate, while stressing the importance of women-only spaces and “dignity” for all.
Penny Mordaunt, meanwhile, published a Twitter thread stressing she has been alongside women “in every major battle”, in response to criticism she is a “woke” candidate.
Mr Shapps told Sky News’ Sophy Ridge On Sunday that he will “not be spending most of my time on these kind of issues” if made Prime Minister.
He added: “When people want to transition gender, that is their choice and they’ll always have the support from me.
“I think the country is far more interested in things like their cost of living, the bread and butter issues, jobs and the rest of it.”
Asked if voters are interested in a so-called “war on woke”, he said: “If people want a PM who’s talking about woke issues at all, it’s just not me, don’t vote for me.
“I am interested in the bread and butter issues that your viewers will be thinking about every single day of the week.
“I am a libertarian, I’m liberal both economically but also socially. Let’s let live. Let people live their lives.
“I just don’t think we need to get caught up in some US-style debate and sort of almost aggressive war on these issues. It’s just not necessary.”
Asked if trans women are women, Mr Tugendhat told Sky: “This is one of those debates that demonstrates why we need to move on, because it is really easy to make division where we need unity, it is really easy to try and divide communities.”
The Tonbridge and Malling MP added: “Look, a woman is an adult human female. But that doesn’t mean in any way that trans women have any less respect or any fewer rights.
Some want to damage my reputation for whatever reason. They want to depict me as ‘woke’. I was elected by the no-nonsense, down-to-earth people of Portsmouth North. It was a Labour seat. I’ve increased my vote share at each election.
I refer you to their judgment.
— Penny Mordaunt (@PennyMordaunt) July 9, 2022
“The idea that you can answer this question with a sledgehammer, when really what you are talking about is a very subtle understanding of people’s rights and people’s dignities and the respect we need for each other, I am afraid is not true.”
He later added he “certainly” believes women’s spaces should be protected, saying: “We need to have a clean start on this debate because actually this debate is about individuals, it is about people.
“In cases like prisons, in cases like refuges, of course there is an answer which is that women absolutely need to have a safe space.”
In a Twitter thread posted before she announced her leadership bid on Sunday, international trade minister Ms Mordaunt stressed she had “stood up for women”.
She added: “I’ve listened to them. I’ve been right alongside them in every major battle.”
Ms Mordaunt claimed some people have tried to “depict” her as “woke” in order to damage her reputation.
The Portsmouth North MP went on: “I’ve fought for women’s rights all my life. I would NEVER do anything to undermine them. I will continue to protect them.
“And those that purport to be champions of women while misrepresenting and undermining them might like to think again.”