The question of the status of women within Islam is (unfortunately) often the victim of highly politicized agendas from both sides.

To speak to this meme in particular, it is true that Fatima al-Fihri founded a madrasa and library in 859 that would later become the University of al Qarawiyyin in Morocco. Such institutions were often established through the waqf system, or system of religious endowments, in which a Muslim man or woman could donate money or his or her inheritance to the maintenance of a property or the running of an institution. Women often donated their fortunes, however large or small, to public waqfs that would provide educational services (such as the madrasa in Qarawiyyin) or would feed the poor. These waqfs were created in perpetuity, until the money to manage them ran out.

This establishment of waqfs by women was not a rare action either. According to Sharīʿa by Wael Hallaq, by the 1800s (obviously much after the founding of the madrasa in Qawayyin), 40 - 60% of all real property in Muslim lands was in the form of a waqf, and women controlled a large percentage of them. Although he doesn't provide statistics for Morocco, women founded 40 - 50% of the waqfs in Aleppo in the 18th century and 25% of the waqfs in Egypt. Thus Fatima would have had the resources, the legal mechanisms, and the social acceptance to establish such a center of learning.

I can't speak to the half of the meme on Western Culture. Obviously cherry-picking examples from either side will give an incomplete view of the picture as a whole and I don't want to try and speak on a matter I don't know anything about. All I'm trying to get at here is that the meme is not unrealistic: the University of Qawayyin did emerge from a madrasa and library established by a woman, Fatima al-Fihri, and it can boast a good claim about being the oldest university in the world.

Naturally, the question of women in Islam is larger than just their founding of charitable foundations. This is a topic I'm pretty interested in, so if you have any other questions don't hesitate to ask either here or in the future in another thread.