The post quickly gained hundreds of comments - with many parents and supporters expressing disappointment - and was deleted about 21:00 GMT.
Commenters accused the club of sexism, called for an apology, and even said they would not be continuing to attend matches at the club going forward.
"As a parent of a Neath Panthers player who has done nothing but support the club, I find this post very disappointing and very insulting to our girls," said one.
Another added: "Disappointed in this 'not for girls' marketing approach. Really shocking considering all those female players in your area."
Prominent sportspeople also took to social media to condemn the post, with Tonia Antoniazzi - Labour MP for Gower and a former Wales international - describing it as "disappointing for a club with such a great history and legacy which has inspired so many women and girls to want to play".
Neath Panthers an all-female Welsh Rugby Union (WRU) hub, aimed at young players aged six to 18, is also based at the club.
Neath RFC club owner Matty Young issued an apology on Tuesday and said he took "full ownership as the club owner and author of the post".
"The idea from the start was to shine a light on the casual misogyny that still exists in parts of rugby and, more importantly, to champion the incredible women and girls' rugby players who represent our club with passion, skill, and dedication," he said.
"I now recognise that the initial post did not effectively convey this message, and for that, I am sorry.
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