Kirk would often have women falling for him and, when it was necessary for drama, it he fell for them, too. Memory Alpha lists romantic relationships of one type or another with 22 different women, with additional ones implied.
They often did not result in any sexual relationship (though not always), and some were old lovers from his past. The original characterization of “space stud” was a joking one – those pointing out knew there was usually no sex involved. But people took the characterization literally.