[Trigger Warning: Domestic and Sexual Assault. Also, BIG comment with many a footnote.]
Here’s an angle I haven’t seen mentioned yet: there’s more than some hidden racism in Dawkin’s focus on Islam, which he exploits to distract from real global problems. The quoted portion in the OP is my starting point:
The greatest threats to women, in his view, are Islamism and jihadism — and his concern over that sometimes leads him to speak off-the-cuff.
“I concentrate my attention on that menace and I confess I occasionally get a little impatient with American women who complain of being inappropriately touched by the water cooler or invited for coffee or something which I think is, by comparison, relatively trivial,” he said.
That’s demonstrably false. Consider, for instance, a 2006 WHO study on domestic violence. It surveyed fifteen sites worldwide, and found that the highest lifetime rate of physical or sexual violence by an intimate partner was 71% in the Ethiopian countryside and 69% in the Peruvian… both of which are Catholic-majority states. Third place is rural Muslim Bangladesh (62%), fourth is rural half Christian, half Muslim Tanzania (56%), fifth was urban Muslim Bangladesh (53%), and so on.
The percentage of women who think being beaten is an acceptable punishment for not completing their housework? About 65% in rural Ethiopia, 45% in rural Peru, 30% in rural Tanzania, 25% in urban Tanzania, and so on. Interestingly, urban Bangladesh women are in the middle of the pack when it comes to saying they should not be beaten (rural Bangladesh… not so much).
How about sexual violence against women that aren’t intimate partners? A staggering 55% of assaulted women in Samoa were attacked by non-partners… and Samoa is majority Christian. In Brazil (majority Christian) that’s 40%, in urban Tanzania it’s 34%, urban Peru it’s 31%, and so on.
The prevalence of injury among ever-abused women ranged from 19% in Ethiopia to 55% in provincial Peru. Injuries were associated with severe physical violence. In Brazil, provincial Peru, Samoa, Serbia [Christian] and Montenegro [Christian], and Thailand [Buddhist] over 20% of ever-injured women reported that they had been injured more than five times.
Although the majority of injuries were classed as minor (bruises, abrasions, cuts, punctures, and bites), in some settings, more serious injuries (broken bones, injuries to ears and eyes) were relatively common. At least 20% of ever-injured women in Namibia [Christian], provincial Peru, Samoa, urban Thailand, and the United Republic of Tanzania reported injuries to the eyes and ears. In Bangladesh, Ethiopia, provincial Peru, and Samoa, over a quarter of ever-injured women reported that they had lost consciousness as a result of partner violence.
It’s obvious that intimate partner violence is a global issue, affecting a huge percentage of women worldwide. However, it isn’t obvious this is tied to religion; Christianity seems to dominate the stats, but that could be because it’s still the dominant global religion, or the researchers couldn’t get into certain Muslim states.
Worse still for Dawkins, Islam is not a monolith. According to a Pew Forum study, while 99% of Afghan Muslims support Sharia law, 8% of Azerbaijanis do. The biggest predictor of support is secularism, not religion; Turkey is 99.8% Muslim, but only 12% of their Muslims support sharia law.
Things get weirder when you look at specific beliefs: 50% of Bangladeshis that support Sharia say that family planning is morally acceptable, while among those that oppose Sharia law… only 28% think it’s acceptable. On the flip side, 28% of pro-Sharia Kazakhs think it’s acceptable, yet 52% of anti-Sharia Kazakhs think it’s fine.
The veil? In Sub-Saharan Africa, only 40% of Muslims agree that women have the right to choose, while in South-East Europe that sits at 88%. How about the ever-popular topic of suicide bombings? 96% of Bosnian Muslims do not think it is justified, or think it’s only fine in rare circumstances, while in Pakistan, the spot with the greatest support, that number sits at 49%. Yes, more Muslims in Pakistan think that suicide bombing is poorly justified than that it sometimes or often is. At least 85% of Muslims endorse non-Muslims practicing their religion freely, with some areas hitting 97%. In South Asia, 76% would be OK with executing apostates; in Central Asia, only 16% are.
And yet, Dawkins is opposed to Islam, full stop. No shades of gray. Every Muslim is an olive-skinned Middle Eastern person that wants every woman stuffed in a burqa. Never-mind the existence of Black Muslims in Nigeria or Caucasian Muslims in the Caucasus, they’re all the same to him. While he loves to toss out the phrase “Islam is not a race,” he certainly treats them as one[5,6] and is happy to exploit xenophobia to distract from more important issues.
You might argue Dawkins wasn’t talking about domestic violence, though, but mild sexual assault and sexism. But that supposes those issues are specific to North America and Europe, and absent everywhere else. Again, that’s just not true; as Hans Rosling loves to point out, there really isn’t much difference between developed and developing countries nowadays, with urban areas of some “developing” countries on par with developed nations.
Just looking at cell phone and internet usage, 45% of Lebanese own a smart phone, 39% of Chileans, and 33% of South Africans; in contrast, 23% of Russians and 21% of Mexicans do. By 2018, it’s forecast that 67% of cell phone users in all of Africa will have a data plan. The fastest internet in the world is in South Korea, the fifth is in Latvia, and the Czech Republic comfortably edges out the United States to take seventh place.
This means that the problems of American Women are fast becoming the problem of Global Women. And as the world becomes more connected, the campaigns and experience of the former can be easily modified and exploited by the latter to improve their lot. The same isn’t true for combating Islam, however; as I pointed out above, that religion is quite heterogeneous and thus you’d have a tough time spreading one area’s fix to another place.
This leaves Dawkins as little more than a wailing, short-sighted bigot.
 “Richard Dawkins Stands by Remarks on Sexism, Pedophilia, Down Syndrome.” Religion News Service. Accessed November 22, 2014.http://www.religionnews.com/2014/11/18/richard-dawkins-stands-remarks-sexism-pedophilia-syndrome/
 García-Moreno, Claudia., London School of Hygiene and Tropical Medicine., Program for Appropriate Technology in Health., World Health Organization., and Women and Health. Department of Gender. WHO Multi-Country Study on Women’s Health and Domestic Violence against Women: Initial Results on Prevalence, Health Outcomes and Women’s Responses. [Geneva, Switzerland]: World Health Organization, 2005.
 “The World’s Muslims: Religion, Politics and Society.” Pew Research Center’s Religion & Public Life Project. Accessed November 22, 2014.
 Well OK, I was a bit shocked when I cracked open my copy of The God Delusion and rediscovered that he endorsed teaching the Qur’an as a source of literary heratige. That bit’s on pages 386 to 387 on my copy, or just look for the last two paragraphs in Chapter 9.
 Malik, Nesrine. “Message to Richard Dawkins: ‘Islam Is Not a Race’ Is a Cop out.” The Guardian, September 20, 2013, sec. Comment is free.http://www.theguardian.com/commentisfree/2013/sep/20/islam-race-richard-dawkins
 Chituc, Vlad. “Islam Isn’t a Race, and so What?” NonProphet Status. Accessed November 22, 2014.http://www.patheos.com/blogs/nonprophetstatus/2014/10/08/islam-isnt-a-race-and-so-what/
 The River of Myths by Hans Rosling | #BillsLetter, 2013.https://www.youtube.com/watch?v=OwII-dwh-bk
 “Emerging Nations Embrace Internet, Mobile Technology.” Pew Research Center’s Global Attitudes Project. Accessed November 22, 2014.http://www.pewglobal.org/2014/02/13/emerging-nations-embrace-internet-mobile-technology/
 “Africa Telecoms Outlook 2014: Maximizing Digital Service Opportunities » Informa Telecoms & Media.” Accessed November 22, 2014.http://files.informatandm.com/uploads/2013/11/Africa_Telecoms_Outlook_Low_resolution.pdf
 “South Korea’s Internet Is About to Be 50 Times Faster Than Yours.” Motherboard. Accessed November 22, 2014.http://motherboard.vice.com/blog/south-koreas-internet-is-about-to-be-50-times-faster-than-yours
[BONUS] Hans Rosling: Religions and Babies, 2012.https://www.youtube.com/watch?v=ezVk1ahRF78