Well this is just silly in how counterfactual it is.
Unlike, for example, the United States and a small handful of other countries, every single European country has paid maternity leave, as is mandated by the Union(1).
Unlike the United States, every or pretty much every country has subsidised childcare options, and increasing the rate of 3-year-olds in childcare and parents' access to it is an explicit policy goal of the EU (2). In some countries, including Germany, parents can sue the town council for neglecting its duty to provide adequate options if they don't get a slot(3).
In fact, every measure to increase female labour participation and decrease the penalty of motherhood in the labour market -- two more explicit policy goals of the EU -- can be construed as a measure to increase fertility. It's not a coincidence that high female labour participation and relatively high birth rates are correlated across developed countries. When women can go back to work more easily after having a child, more will decide to become mothers and more will be working (all else equal) (4).
Short summary: It's not "the EU", and least of all leftist policies, that makes European have few children. If anything, conservative, implicit or explicit "women back to the kitchen sink" ideologies stop them from having more.
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(1) See for example here:
http://www.europarl.europa.eu/RegData/etudes/ATAG/2016/593543/EPRS_ATA(2016)593543_EN.pdf
(2) In many countries in Western Europe, the rate is at 90%+ already for 3-year-olds. A country-by-country breakdown of the actual participation rates in pre-primary education of 3-, 4-, and 5-year-olds can be found on page 64/66 of this document:
http://eacea.ec.europa.eu/education/eurydice/documents/key_data_series/166EN.pdf
Here's a comparison of the EU's gross rate with that of the US:
https://data.worldbank.org/indicator/SE.PRE.ENRR?locations=US-EU&view=chart
(3) Source in German:
http://www.deutscher-familienverban...184-rechtsanspruch-kitaplatz-fragen-antworten
(4)
http://www.europarl.europa.eu/RegData/etudes/IDAN/2017/614646/EPRS_IDA(2017)614646_EN.pdf - a quote from page 14:
"Fertility rates are falling worldwide and are associated with growing economic
and social development. However, research suggests that once a certain level of
development is achieved, fertility rates may stabilise or recover to some extent. The
interactions between policies to support families and diverging fertility rates do not
suggest clear solutions, although a common characteristic among countries with stable
or even increasing birth rates is a high degree of female labour force participation."