Are women required to wear skirts to get into some tourist sites?

Skirts specifically are not required, but modest dress certainly is, mostly in religious sites: churches like St. Peter's in the Vatican, major temples in Thailand and India, mosques pretty much anywhere if they're even open to visitors, etc. From the Vatican's official site:

Access to Vatican Museums, Sistine Chapel, Vatican Gardens and Saint Peter's Basilica is permitted only to visitors dressed appropriately (no sleeveless blouses, no miniskirts, no shorts, no hats allowed).

You can improvise though, eg. many temples in Thailand will loan/sell you a sarong wraparound for your legs.

As @hippietrail said in a comment, I know by experience that Orthodox monasteries require girls and women to wear long skirts to enter the building (I visited some of these monasteries on a school trip).

Knowing that before leaving, I brought with me an old skirt and put it on over my jeans just before entering, but the monks had many skirts to be used the way I did.

This is the only case I know that requires women to wear a skirt, while other religious sites that I have visited (mainly catholic churches) have a more general "proper clothing" dress code.

There are similar answers, but mine is more about Asian sights.

If you are visiting Buddhist places, temples do not really force male or female to wear specific clothing, but too revealing cloths are frowned upon by locals. If you plan to visit a Buddhist place, try not to wear shorts, bikini, etc. Anything that covers until your knees and covers shoulders are generally accepted. Also, as for Buddhists, people wear white-ish clothing to temples. Try to wear something of white color, but not red, black, etc.

You can easily find something to cover you up outside of such temples.

Also, be careful when you wear T-Shirts with a religious leaders, icons, etc printed. (I'm from Sri Lanka, where it's an offense to print such images on fashion items and inappropriate materiel).

If you are visiting an Islamic place, a mosque for example, sometimes there are rules (rather than customs). The national mosque in Malaysia, for example, allows women to visit then place, skirts are allowed, but if the dress is too revealing, they provide you some clothing to cover you (free, you have to return it).

Update: Thanks to @Ida for mentioning this: Some Islamic places require covering your head/hair if you are a woman. I would appreciate if someone can help me with the wording, but depending on the culture, you might be asked to wear a Hijab, Burka, or something like that to cover your head as well.

For Hindu places, a dress code is hardly necessary, but generally you wear something that covers your knees and shoulders. If you are too into the Hindu places, do a little research about the corresponding god (most Hindu religious places are dedicated to a specific god: Vishnu, Krishna, etc), do some research about what are the colors relevant to that god. You will be highly appreciated if you wear that color to the place. This is not about obeying their customs, but about mixing up with the locals. One time, I was invited to take part in a dance simply because my T-shirt color matched what they were wearing, and I showed some interest.

Others have mentioned about Christian places, so I'm not going to mention it here.

For other sights, such as diving (say iFlySingapore), skirts are not practical. Just use your common sense and try to wear appropriate clothing.

Depends a lot on the location, country, culture and religion in question. Most countries have their own ethics and morals, which often encompass dressing styles and restrictions.

Religious locations tend to be one of the most common places were a "dress code" is suggested. And often revealing or sexy clothing is not appropriate to wear.

As travelers we bear the responsibility to learn about our destination, so as to be able to visit their country as polite guests, not disrespectful louts.

As you mentioned the Vatican, women are not restricted to skirts only, pants and shorts are both acceptable, however shorts should come down below the knees and no sleeveless tops, shoulder need to be covered. But this applies primarily to going inside the churches or chapels.

Yes, for example most monasteries in Georgia, including Gergeti Monastery, require women to wear a skirt. This is not a big deal though, you don't have to come prepared, as long as you're dressed modestly. This includes not wearing open shoes or sandals for instance. Apart from that, there is a huge bunch of skirts right next to the entrance of those monasteries where wearing a skirt is compulsory, and women can just put it on their clothing/trousers. Even if it looks slightly ridiculous, it doesn't really matter for the few minutes you're spending in the church. Naturally, borrowing a skirt at the entrance is free and there's nobody standing next to it checking who's wearing what. If you do enter the church without a skirt (same goes for men with a hat on), the priest or whoever is the responsible guy in the church at that moment, warns you or your tourist guide politely to fix that mistake.

From all the places I've been to the only ones which require women to wear long skirts were Eastern Orthodox Churches and Eastern Orthodox monasteries. Those places also require men to wear long pants, and no shorts of any kind were allowed.

Any other religious places - mosques, Buddhist and Hindu temples, as well as other Christian Churches do not have such requirements. Many of those places prohibit sleeveless shirts/tank tops for both genders, and prohibit skirts/shorts well above knee (for women this seem to be only enforced with bare legs, i.e. a short skirt with tights is ok; short skirts/shorts for kids is also ok). However even in Vatican - and I just been there last Sunday - there is no requirement for women to wear skirts.

I almost always wear long shorts, and Eastern Orthodox churches (around ten of them) were the only ones I wasn't allowed in.