The "Make Noise for Free Choice" campaign is timed to coincide with the upcoming EU parliamentary elections in June and has been initiated by Liberal Women - a group within Sweden's Liberal party and headed by Birgitta Ohlsson.
"Our aim is to have one million signatures by October 2010," Ohlsson told The Local.
The initiative is supported by Baroness Sarah Ludford, Liberal democrat Member of the European Parliament (MEP) for London, Wanda Nowicka, President of the Polish Federation for Women and Family Planning and Sophie in 't Veld, a Dutch MEP.
"When a state denies a woman the right to her own body, integrity and health, it is a violation of her human rights," according to a statement on the campaign's homepage.
The initiative argues that the freedom to choose to have an abortion should be enshrined with the other established human rights as a criterion for EU membership and no longer be considered a mere health issue, sorted under the principle of subsidiarity.
"When there are safe abortion methods no woman in the EU should have to go to a quacksalver to terminate an unwanted pregnancy. It is a scandal that this still occurs in Europe in 2009," Birgitta Ohlsson writes.
"Women who lack the access to free, legal and safe abortions in the EU-countries such as Ireland, Malta and Poland, can no longer be ignored."
Ohlsson told The Local she was expecting to face opposition from certain quarters, including the Catholic church.
"There has been a conservative backlash in recent years. In Poland, for example, abortion used to be legal. What I find is that we often have the majority of people behind us but the conservative leaders have become more aggressive," she said.