As Oliver pointed out, one of the problems with the gender wage gap is how to quantify it. Different studies lead to different results. But no-one actually thinks that there is no gender wage gap (which explains tweets like this one.)
Even Christina Hoff Summers, who some feminists would consider to be a men's rights activist/apologist, does not say that the gender wage gap does not exist. Rather, she claims that most of it can be explained away by life choices and different work—exactly the views mentioned by Oliver— leaving a smaller 6.6 cent difference which could reduced "to a few cents" with "more realistic categories and definitions". If not even Hoff Sommers can bring herself to deny that there is a gap, but only say that the gap is smaller than most people think, then one thing is clear: the gender pay gap exists.
By the by, the gender pay gap deniers always put me in mind of the famous cartoon by Sidney Harris:
The next thing to mention is that, even though there is discussion about how big the gender pay gap is, to deny it exists you would have to question why pretty much every government in the world publishes statistics confirming that it exists. What, exactly, do governments stand to gain by stating that half of their populations earn less than the other half? (If you have to posit a feminist takeover of governments all over the world when the UK and Germany have had one female political head of state each, and the USA none, then see the above cartoon again.)
But, just for a moment, let's take the 'life choices' argument seriously. Women are paid less because of their life choices: having children, staying at home to look after them, and so on. Fine. Let's take childbirth. Only women are able to have children; this takes them out of the workforce for a certain period of time, and afterwards they have to play 'catch-up' on the men and women who did not have children. Economically, having children makes you a less desirable employee for a company, and this is reflected in lower wages.
There's two things I'd like to point out with this. First of all, until it was abandoned in 2011, young German men had to spend a year in military conscription, with an option to work in civil services (such as healthcare) instead. One of the most commonly citied reasons for young German women not having to do conscription was that they had to have children at some point in their lives. Regardless of whether this 'reason' was official or not, the fact remains: German men took time out from work for conscription; women took time out for childbirth. Did this affect the gender pay gap? Not. At. All.
If you're going to maintain that the gender pay gap can be explained away by life choices, you need to explain why women's life choices seem to impact the gap more than men's life choices—even if those choice are made for them by the government.
But there's another question that needs to be answered as well. While there might be a number of different ways of quantifying the gender pay gap, which implies that the are different ways of looking at one economy, it is still possible to look at different economies in the same way.
The graph below shows the gender pay gap across the 28 countries of the EU in 2013:
According to this data, the average gender pay gap across the EU is 16.4%; the gap in Estonia is 29.9%, and the gap in Slovenia is 3.2%. Note that Germany (21.6%) and the UK (19.7%) are far worse than the average.
The point that I'm trying to make is this: even if you think that 'life choices' account for the difference between the pay of men and women, you still need to explain why there is such a difference in the gender pay gap between these countries. After all, women all over the world have children. Taking the most extreme examples of the pay gap above, the fertility rate in Slovenia is 1.58: in Estonia it is 1.56. In Germany, the fertility rate is 1.28; in the UK it is 1.92.
The fertility rate in Estonia and Slovenia is almost identical: yet the gender pay gap differs by 26.7%. Now of course, having children is not the only 'life choice' women make, although it is the most obvious and most widely cited. But if 'life choices' mean anything in this discussion, you need to explain why Slovenian women are either not as affected by those life choices, or why they are not making the same life choices than Estonian women are.
No matter whether you believe that the gender pay gap can be explained away by life choices or not, you still need to account for the difference between countries.
And if you think that life choices account for the pay gap, you still need to explain how, and why, they are so different across the EU.
'Life choices' ultimately explain nothing: they just give you something new to explain.