America is the most populated, influential, and powerful Western country whereas Saudi-Arabia follows a version of Islam that accounts for less than 1% of the world's total Muslims and also has a similarly tiny population (with disproportionate influence due to Western money).

Not exactly an equal comparison.

And if you're picking Saudi-Arabia because of its location (Arabia), then you should know the following:

Before the Saudis, Arabia (the part with the holy cities at least and Palestine) up until WW1 was ruled by Turkey which was at the time the Islamic version of the Vatican. They, the Muslim theocratic government which was the central authority of Islam, were kicked out of Arabia by the Saudis (who answered to the British through Lawrence of Arabia... simplifying it but that's the gist of it). The British and Americans gave these kingdoms to the Saudis and current Arab rulers.

Less than 100 years later and the Caliphate ceases to exist and Turkey has banned the hijab from government buildings (it's still popular with the people though).

The Vatican is still around in Rome and treated like a sovereign nation by the West.

Who's more free from religion? The guy you're disagreeing with, who has personal experience in the matter (and others like him), are saying this is ingrained into the culture which means it cannot be a simple matter of men telling women to put it on. The women are convinced they want to wear it much like how women in the West are convinced they want to wear high heels and other strange things because men have taught them to. That is what the picture is expressing.

Who popularized the bikini? Well it emerged initially for women's swimming (first from competition, then for swimming in general), but it became a cultural phenomenon seen on beaches everywhere due to men (in Hollywood).

Hollywood endorsed the new glamour with films such as Neptune's Daughter in which Esther Williams wore provocatively named costumes such as "Double Entendre" and "Honey Child".[17] With new materials like latex and nylon, by 1934 the swimsuit started hugging the body and had shoulder straps to lower for tanning.[18]

[...]

Finally, the modern bikini was introduced by French engineer Louis Réard and fashion designer Jacques Heim in Paris in 1946.

Regarding high heels (which originated as dress for royalty in Europe and for horse riders elsewhere),

Throughout most of the 19th century, flat shoes and sandals were usual for both sexes, but the heel resurfaced in fashion during the late 19th century, almost exclusively among women.

Women's fashion was still run by men in the 19th century. More:

Reasons against wearing high heels, which are almost exclusively health and practicality reasons, include:

[...]

Reasons for wearing high heels, which are almost exclusively aesthetic reasons, include:

[...]

In modern society, high-heeled shoes are a part of women's fashion, perhaps more as a sexual prop. High-heels force the body to tilt, emphasizing the buttocks and breasts, highlights of a woman's sexuality. They also emphasize the role of feet in sexuality, and the act of putting on stockings or high-heels is often seen as an erotic act.[9] This desire to look sexy and erotic continues to drive women to wear high-heeled shoes, despite causing significant pain in the ball of the foot, or bunions or corns, or Hammer toe. A survey conducted by the American Podiatric Medical Association showed some 42% of women admitted that they would wear a shoe they liked even if it gave them discomfort.[10]

Lastly,

www.thedailyshow.com/watch/wed-july-1-2009/burka-ban