Thanks for the interesting question! This response is going to focus on the 1900s and 1910s, a little earlier than the Negro Motorist Green Book; and on American urban and suburban society, mostly in the northeast, as opposed to what I understand to be the more rural and southern focus of the Green Book (unfortunately I have not read it).

My direct answer to your question would be that white women motorists certainly faced discrimination, but it was not the legal and mortal danger that black motorists risked. White women's access to automobility was, in the right circumstances, pretty much limitless, and forceful advocates for women motorists arose at a very early stage. However, misogyny, social and familial pressure, and the substantial cost of early automobiles meant that as late as 1917, only 8% of motorists in Massachusetts and 23% in Southern California (the highest percentage in the US) were women.

To look at how automobility became an option for women at all, we need to take a look at the late Victorian period. The last decades of the 19th century had seen a significant shift in women's role in public spaces - from the workplace, to the marketplace as (even more) independent consumers, and, in a literal sense, in public spaces such as parks, recreation centers, and the street. Women's access to automobility in the early 20th century, such as it was, was influenced heavily by another mobility technology from that era - the bicycle. During the "bicycle craze" of the 1890s, as much as a quarter of all cyclists were women. On bicycles, women were able to successfully assert their right to personal mobility, independence (even if many columnists suggested that women cyclists be accompanied by chaperones), and right to use the street on an equal footing with men. As we will see with the automobile, however, strong social pressures still existed: women cyclists were expected to dress modestly, avoid cycling at high speeds, and maintain an appropriate posture. Women were expected to purchase bicycles specifically "tailored" to female riders - with low crossbars, wide seats, and an emphasis on grace and lightness rather than speed or durability; they were expected to choose bicycles with short cranks, in the belief that long cranks which raised the knee above the hip led to “unhygienic” contortions. Evan Friss argues in The Cycling City that what began (with the penny-farthing "highweelers") as a strictly masculine pursuit was transformed by the end of the 1890s into a means for both men and women to express their masculinity and femininity. These were social expectations, of course - those with an abundance of social capital, or who were not interested in their social standing, were free to do as they pleased.

While the bicycle set the stage in a number of important ways, aspiring women motorists still faced significant - and in some respects, more significant - obstacles to full social acceptance. While there was never a legal restriction on women motorists anywhere in the US (to my knowledge), they were certainly considered - in 1908 in Cincinnati, after a woman motorist ran over the city treasurer, the Mayor proposed a ban. In 1909, the prominent Anti-Feminist Montgomery Rollins made his case for a ban in the popular magazine Outlook. Although he acknowledged that female motorists had half the rate of accidents as men, he suggested that men who broke traffic laws be excluded from the statistics, since, (I kid you not), "the courts will punish them anyways." He rested his argument, then, on his unsupported assertion that, when considering only drivers who are obeying the law, women's accident rates exceed that of men; and that the majority of women were not strong enough to control their machines, the only thing to do would be to ban all of them. None of these bans came to pass - at the very least, the nascent automobile lobby was strong enough to prevent the loss of any potential market, even one which it disregarded and demeaned. Private organizations were not so easily swayed, however - the Automobile Club of America in NYC, which had the best garage in the city for many years, banned women from membership; and in 1909, after a woman (pioneering auto racer Joan Cuneo) defeated the legendary Ralph DePalma (winner of the 1915 Indy 500) in a race in New Orleans, the American Automobile Association banned women from racing.

A number of historians have argued that the dramatic industrialization of the late 19th century sparked a change in how men defined masculinity. As more and more men took factory and office jobs, traditionally "masculine" attributes such as aggression, independence, and physical prowess, became harder to assert. As a result, men focused on a professional and competence-based masculinity. This involved, among other things, claiming an innate intellectual superiority over women; and, more specific to this answer, began to assert mechanical ability as a masculine virtue. Where a bicycle could be reasonably be asserted as within a woman's competence (and indeed a number of by-women-for-women bicycle repair manuals were published during the boom), the substantially more complex mechanical workings of the gas automobile landed it squarely in the realm of a masculine status object. To borrow an argument from Gijs Mom in The Electric Vehicle, early automobiles were as much an "adventure machine" as a practical means of conveyance, and for early male motorists, the maintenance and repair was part of the adventure - a way to assert mastery over a machine in a time when many men felt subservient to the machines of their employers. This assertion of mechanical ability as a masculine trait necessarily excluded women, particularly for gas autos: it is not a coincidence, then, that manufacturers of electric vehicles (which have far fewer moving parts) decided to market their passenger vehicles almost exclusively to women. Some of the ways in which this exclusionary attitude presented itself will probably not sound too foreign - a "humorous" article published in 1916 recounting an incident when some male mechanics convinced a woman driver to oil her spark plugs (causing them to immediately foul) - a woman motorist who took great pains to perform any necessary repairs in deserted locations, to avoid "snide comments from male kibitzers" - a woman who complained that traffic policemen would jump out of the road in mock terror when she approached an intersection - and dozens on dozens of stories of women motorists receiving delberately misleading answers when stopping for directions.

In addition to the casual misogyny and harassment women motorists recieved from the public, they faced significant pressures from their families. Becasue of the expense of automobiles, women almost always were driving cars purchased for them by husbands or fathers: according to New Hampshire registration data from 1911, of the 4.8% of automobiles registered to women, the majority came from addresses with a male registrant of the same name. Most of the earliest women pioneer motorists had to get permission from their fathers or husbands (a notable exception being Betha Benz, who took her husband's newly invented gasoline automobile on a trip from Mannheim to Pforzheim to drum up publicity for the machine, without telling him). President Roosevelt, for example, repeatedly refused his daughter permission to buy an automobile, and the two quarreled publicly when she borrowed a friend's car.

I hope this answers your question, and I hope it makes sense - I'm not sure that I've laid it out in the best way. If you have follow up questions, I'd be happy to answer.

Sources:

Clay McShane, Down the Asphalt Path: The Automobile and the American City
Evan Friss, The Cycling City: Bicycles and Urban America in the 1890s
Gijs Mom, The Electric Vehicle: Technology and Expectations in the Automobile Age