That article is pretty lacking. For its length, it employs a lot of fuzzy logic, lacks proper context and doesn't support its own points. I'll go through the more well supported points. I'm going to have to do a bit of reorganizing because the article is basically a gish gallop and poorly structured:
1. "Then you make a giant leap from group differences between men and women on such measures as interest in people rather than things, or systematising versus empathising, to differences in men’s and women’s ability to code"
The author seems to think that ability to code is some inborn talent that must be evenly distributed between men and women. Coding ability (like math, knitting, doctoring and literally every other human skill) is a confluence of natural talent, interest, coaching, and raw time spent. People can have equal talent for a skill, but ultimately have radically different ability due to their chosen paths in life. If Damore's hypothesis is correct and men do have a greater natural interest in coding, then we could expect men to have a greater ability to code, on average, even if men as a whole had less aptitude for coding. Damore didn't clarify because he probably didn't expect anyone to interpret his words with such malice.
2. "You ignore many other gender differences, basing your argument only on a few that you think support your conclusion."
These differences should be cited and used to build a counter argument.
3. "You’re ignoring everything else that could explain the gender gap."
Damore's memo does not take place in a vacuum. It's an alternate view offered in contrast to the popular mainstream view.
4. "The gender differences you cite differ between countries and over time."
Not an argument, or even a rebuttal. More of a confounding factor.
5. "They (gender differences) don’t even support your argument, because you don’t seem to understand what makes a great software engineer...Software engineering requires a broad mix of skills ... Teamwork, in particular... Senior engineers must manage teams—and by your own reasoning that should mean that women, with their greater empathy and interest in people, should be over-represented at that level."
Being a low level software engineer is a prerequisite to being a high level software engineer. If women are underrepresented at the bottom, it's expected they will be underrepresented at the top. You might expect an "enrichment" effect of women in management, but there may be other factors (that I'll talk about later) that result in women leaving the software profession before that point. All of this assumes that high empathy and interest in people are even required to be a good manager. Steve Jobs was often considered a good leader but it doesn't seem like he gave a crap about the people under him.
6. "You clearly don’t understand our company, and so fail to understand what we are trying to do when we hire"
Not an argument.
7. "Even if you are right that more men than women are well-suited to the job of software engineer at Google, you are wrong that taking steps to recruit more women is inherently unfair to men."
Not an argument.
8. "There is plenty of evidence that women in Silicon Valley suffer sexism and discrimination. Read Susan Fowler’s description of the harassment she experienced at Uber before leaving the firm in December. Look at the responses to “Elephant in the Valley”, a recent survey of senior women in tech: among its findings was that two-thirds had been excluded from networking opportunities because of their sex. And beyond our industry, women are less likely than men to be given plum assignments, are given less useful feedback, are seen as pushy when they ask for pay rises (men are seen as ambitious) and, in leadership roles, may be seen as either competent or likeable, but rarely both. “We need to stop assuming that gender gaps imply sexism,” you write. But we know there is sexism! We don’t need to infer it from the existence of gender gaps."
An argument! All of this is certainly true and should be addressed by internal and legal processes. In addition, all of these are potential explanations of the gender gap in software development. However, these points alone are not sufficient to make that argument. What we need here is context. Does the rate of discrimination differ between fields that are lopsided, like computer science, and fields that are more even, like law and medicine?
9. "Finally, let’s look at your contention that by trying to recruit more women Google is discriminating against men. You seem to think that if we stopped paying any attention to applicants’ gender when deciding who to hire, we would naturally converge on the “right” share of men and women, that is, the share that matches the distribution of talent in the recruitment pool. I don’t believe that. Unless we make special efforts, some women will be put off applying by the heavily male culture; those doing the hiring will be influenced in their assessments of candidates’ ability by the stereotypes they’ve formed in that male environment. We should “treat people as individuals, not as just another member of their group,” you write. That is what we are doing."
This is so much feeble it hard to figure out where to start. First, no one really cares about talent in the recruitment pool, even if they use the word "talent". What they really care about is ability (see my above definition of ability). If you have solid math skills but can't tell a for-loop from an if-statement, google probably isn't trying to recruit you for a programming job. "Special Efforts" is the problem, not the solution. Harvard makes "special efforts" not to recruit asians. Fifty years ago, companies made "special efforts" to hire white men. "Special Efforts" are discrimination, full stop. Reverse sexism is still sexism.
In this article's quest for "context", they missed all of it. There are a couple of major data points that they glossed over.
1. The paradox of gender ratios in STEM fields by country. Gender ratios in STEM fields are most extreme in countries that are most gender equitable (like the nordic countries) and least extreme in countries known for sexism (software developers in Zimbabwe are 50% women and india has a much higher percentage of women in programming). One explanation is that Software development in these countries is composed of more westernized individuals who have more progressive views than other people in their country. The other explanation is that talented women are basically "funneled" into computer science in those countries because it's even harder for them to enter other professions.
2. The prevalence of women in law and medicine. The stigmas against women entering these was pretty extreme, yet women smashed these barriers to the point where most people entering the medical profession are now women. There are still areas in medicine that are male dominated, but there are also areas that are female dominated.
There's also a philosophical question that needs to be addressed. Is the existence of a gender gap proof of sexism? If yes the forensic science and veteranarian science have a lot to answer for. Those professions are more lopsided in favor of women then computer science is in favor of men.
Obviously the issue is complicated, but it's not clear that the discrimination that women face in computer science is greater than they face in other fields where representation is more equitable. My biggest takeaway from Damore's memo was that Google was 110% committed to the idea that the gender imbalance is due to sexism. And since it can't find particular instances of sexism to punish or stop, everyone must be so guilty that they can't see it. Google's response to the Damore memo says quite a lot.