What would you argue was the most critical ideology in 19th century Europe?

A strong case can be made that Marxist ideology played a critical role in shaping European culture, society, and politics.

There can be a variety of answers to an open-ended question like this one. I believe that Marxist ideology played a vital role in shaping institutions in 19th century Europe because it caused people to reconsider capitalism.  As the 1800s began, capitalism was on the rise. Nations like England had taken to the factory system, clearly marking the shift from governmental Mercantilism and the cottage industry philosophy of economy. Governments started to embrace an approach that favored liberal economic growth. As a result of capitalist expansion, urbanization began to take hold of European societies. Cities became a consolidation of wealth and poverty, promise and blight. The very rich were set apart from the majority of the poor, as the class structure became an embedded part of 19th century European society. 

Marxist ideology insisted upon examining these social, political, and economic conditions. Marxist thought was radical for its time on many levels. One reason was because it represented a strong example of philosophical "praxis," a meeting place between theory and reality. Marx was not content with being confined to philosophical discourse; he was direct in suggesting the social dimension to his thought. He unabashedly offered it as a scathing critique on capitalism. He took thinkers like Hegel and Feuerbach and used them in his ideology to speak to modern times. When Marx argued alienated labor was intrinsic to a capitalist system, he was not merely addressing a philosophical form of consciousness; he spoke to the reality of countless workers across Europe. The Marxist ideology's ability to speak for an entire class of people was one way it was the most critical perspective of the 19th century.

Marxist ideology insisted upon change. When Marx and Engels write that workers have "nothing to lose but their chains" and call upon them to "unite," it is a call to action. It is an ideological insistence to change what is and transform it into what can be. Marxist ideology insisted on viewing daily life as an extension of capitalist practices, demanding that it can be changed. Marx's philosophical emphasis on the stages of economic production enhanced this with its prediction of capitalism's demise, inspiring great fear among the wealthy. Marxist ideology's demand for change can be considered another reason why it was the most critical set of ideals in the 19th century.

Even if we concede that Marxist ideology might be limited, the ideology's critique of capitalism was quite compelling in 19th century Europe. Massive proliferation of wealth for the few alongside poverty for the many caused significant discussion and action. Marxist ideology forced people to reconsider the world and their places in it. Governmental suppression of Marxist thought along with the social upheaval it inspired reflect the pervasive effect that Marxist ideology had on 19th century European institutions.