Holy crap, this thread is a desolate wasteland of baditutde. First of all, absolutely no terms have been defined whatsoever.

In what sense are terms like "excel" and "advanced" meant? Technologically? Culturally? Militarily? Some other sense?

In what time period are we referencing? Greek culture has lasted, in one form or another, fairly continuously from about the 8th Century BC to today. The Romans (not counting the Kingdom of Rome) lasted from the 5th Century BC to the 5th Century AD (if you don't count the Byzantines). Even if you just mean "Ancient Greece," you're talking about a period of more than a thousand years.

The Indians and Chinese were kinda sorta there through all of that, although they didn't exactly exist as such, and were largely a collection of smaller states that lacked a single cultural identity and spoke a whole bunch of different languages. I'm not as familiar with India, but in China, you had everything from the Zhou all the way through the Warring States and the Qin (the first real glimpse of "China" as we think of it today) and the Han period, almost into the Tang Dynasty as this was going on. It wasn't really until the Han, though, that China had a solid grip on its cultural identity and had entered into a period of piece (at least in the Western Han) that allowed its technology and art/philosophy/literature to flourish. This period roughly coincided with the height of Rome, as well. As the Roman Empire began to collapse, the Chinese entered into a period of civil wars (called the Wu Hu period -- easy to mnemoically remember because it sounds like "boo hoo" and lots of people were crying, my Chinese history prof taught me that), and didn't emerge back into another age of peace and achievement until the Tang Dynasty, by which point the Western Roman Empire had long since collapsed and Europe had gone into the Dark Ages.

So, again -- what the heck time period are we talking about here? There's more than a thousand years to compare!

All that said, I want to address this:

are only studied because of their whiteness. How accurate is this?

This is, frankly, a bunch of absolute crap, and anyone who said this to you cannot be a serious student of history. Greece and Rome are not studied "because of their whiteness," they are studied because they are two of the most influential cultures, in nearly every possible sense, in world history. This does not mean that China and India are not, likewise, highly influential. Just to illustrate, briefly, some of this equivalency:

Again, I can't speak much about India, but during this period, two of the world's most important religions (Hinduism and Buddhism) arose from there, and in China, Confucianism and Taoism were taking hold. Likewise, the world's most popular and influential religion, Christianity, began to split off from Judaism in the Roman Empire during this period.

At its height, the Roman Empire comprised perhaps 20% of the world's entire population (90-100 million people), and stretched from Egypt to the north of England, where Emperor Hadrian built his famous wall. In China, the legendary Emperor Qin Shi Huang began work on his own Great Wall, and a few centuries later, by the time of the Tang Dynasty, the population had reached nearly 50 million people, the largest contiguous population group on earth.

The Chinese military strategist and philosopher Sun Tzu was active during this period, and greatly influenced the strategies and methodologies of Chinese militaries of his time -- and all sorts of people, military and otherwise, down to this day. About a century later, the military genius Epaminondas of Thebes was one of the first people ever to discover and apply true battlefield tactics in the modern sense (in particular the idea of localized numerical superiority) at the Battle of Leuctra to crush a Spartan army thought to vastly outmatch his own.

Lao Tzu, the famous Taoist philosopher and mystic and poet, propounded his ideas of free thought, individualism, going-with-the-flow, peace, etc., that continue to inform modern philosophers, during this period. Around the same time, Socrates, Plato, and, later, Aristotle and others, laid the groundwork for pretty much all of worldwide philosophy whatsoever, whether it be ethics, politics, aesthetics, epistemology, metaphysics, etc., having an absolutely incalculable effect on the shape of the world.

There are some areas where it is difficult to draw parallels. The Greeks and especially the Romans appear to have totally surpassed their Eastern counterparts in the fields of art and architecture, having understood concepts such as linear perspective, sculpture in the round, concrete, arches, complex engineering, gears and pulleys, drama and theater, acoustics, rhetoric, epistolary, and many other art forms in ways that the Chinese (and Indians to my knowledge, I add this with reservations as I am not an expert on India) do not seem to have understood at this time -- at least not on a large scale. Perhaps most important, however, was the Romans' language itself -- Latin -- and its concomitant alphabet, which we are using to read and write this now, and which have had, together, perhaps the largest effect on world culture of any human creation of any kind whatsoever.

At any rate, I don't know if this answered your question at all -- I started to ramble. All cultures, especially major ones like the Greeks, Romans, Indians and Chinese, are certainly worth studying, and beware anyone who tells you that something is "only studied because of x, y, or z political reason."

EDIT: To be clear:this latter part of my comment is a defense of the study of Greece and Rome as equally worthy of the study of India and China (and anything else); it may therefore appear to be biased for the study of Greece and Rome -- but that is exactly the point, it is responding to a criticism of the study of Greece and Rome as a mere cultural bias rather than something objectively useful. Accusing me, therefore, of cultural bias for responding to accusations of cultural bias seems... strange.

People living in Asia tend to study more Asian history, people living in Europe or in areas dominated by European settlement tend to study more European history. Really there is no need to invoke a racial conspiracy theory or some cultural dick swinging contest here. Proper historians and anthropologists simply don't talk in terms of one civilisation being 'more advanced' than another, they look at individual technologies and organisational structures and how cultures adapted to their particular circumstances. If a forest dwelling people surrounded by hogs and fruit trees and roots don't develop agriculture, it doesn't follow they aren't advanced enough, it can simply mean they have enough food readily available that they have no need of that technology, similarly people living in cold climes have little use for air conditioning, and there's numerous other such examples.. so to pick out individual technologies as somehow being markers of progress is meaningless, because success of a civilisation, as with a species, is about ability to adapt, not ability to get all the techs as listed in Sid Meirs' Civilisation Games. /rant.

The simple answer: yes and no
All joking aside I'm gonna break your comment down into two parts. Part one Greeks and Romans vs Chinese and Indians. Part two are they only studied because of their whiteness.
I'm gonna based how advanced they were based on their tech because its harder to dispute than culture. Note i am comparing them at about 220 AD.
The Chinese at this time had mechanical devices like air conditioning (aka a very good ventilation system that can cool a room) and machines that separated wheat from the chaff. On the other hand Romans had perfected concrete (it fell out of use until the mid 18th century when it was "re-pioneered") and also building domes.
The Romans and the Chinese were both advanced civilizations but they excelled in different things. This is the same of Greeks and the Indians. Was one more advanced than the other? Not unless you go into specific areas, then one or the other will excel but in general they were both advanced.
Now to part two. Why do we learn about them so much. This is much simpler. Ask yourself this where do you live? Who was it first colonized/settled/ruled by? Were these colonizers/settlers/rulers once ruled by the Romans? If you answered yes to that last question than the Romans probably imparted some of their culture onto them and they taught about them when they colonized/setled/ruled that area of land. Also the Romans thought highly of the Greeks and Greek culture was incorporated into roman tradition and customs. That is why they are taught about in your land.
If you have any questions regarding sources for specific details please comment and I will be happy to give them, i am on a bit of a crappy computer and even just typing this out is challenging.
Edit:Just giving a term a more relevant name.

What would make anyone think Rome and Greece were "more advanced" than the Chinese or Indians? I thought we had gotten around that phase of Eurocentrism.

To answer your question, the post you're talking about sounds like one of those this western civilization vs. that eastern civilization flame posts. While enraging, it's no more stupid and horrible than the generic her der the west is way more advanced than the east stupidity.

Civilizations excel in their own way and have different societies as well as cultural outlooks.

Romans pioneered in many things like concrete constructions, aqueducts, arched structures, newspapers, etc.

On the other hand China had paper, silk, tea, gunpowder, the compass, the Art of War by Sun Zi etc.

The development of individual societies are not, and never will be symmetrical or parallel. Everyone builds off the foundation of predecessors.