According to their own survey responses, anti-abortion voters are hostile to gender equality in practically every aspect.
The Catholic church, an unabashedly misogynist institution that to this day refuses to allow women into positions of power, had long opposed abortion (but not for all that long – until about 150 years ago, the Catholic view was that abortion was permissible through the first few months of pregnancy).
The birth control pill brought with it an avalanche of opportunities and freedoms, and women, finally fully able to have sex for fun and prevent pregnancy, took full advantage. The ability to delay a pregnancy – and later, the ability to legally end one – meant that women didn’t have to choose between romance and ambition (and it meant women could be choosier about romance, making a more considered decision about who and whether to marry).
This undermined the whole rightwing Christian project, which was, and remains, thoroughly invested in a nuclear family with a father at the head. And indeed, rightwing arguments against abortion used to invoke conservative gender tropes much more often – that abortion undermined the traditional family, for example.