I would probably object to the word "civilized." Anyone that actively seeks to profit off of the enslavement of others and pillaging of nations which are not theirs loses the title of "civilized" in my calculations. This is not to say that European nations possessed multiple advantages in the relationship between themselves and the nations they controlled. Certainly, they benefited from increased industry and greater economic and material wealth. This helped in the race to and process of colonize the world over. Additionally, advances in science, technology, research, as well as the general mode of thought that would allow European nations to exert a great deal of worldwide control and influence in the process of colonization and imperialism.
Certainly, the Europeans were more civilized and more developed. This is why they were so much stronger than the natives of the places they colonized. If they had not been so much stronger, they would not have been able to colonize.
Now, does that mean they were better in some way than the natives? Not necessarily. It just means that, for a variety of reasons, they had more technology and, generally, a more organized and centralized system of government than the people they conquered.
So, there is no denying that Europeans were more developed. If development and civilization are the same thing, then the Europeans were more civilized too.
This question can be answered only with respect to only a specific period in history. At different times, people in different parts of the world have been more civilized than the others. Also there is a difficulty in defining the term civilizations. If we use the word civilization to mean extent to which people have adopted a life style making greater use of technology, things are relatively easier. But people some times tend to define civilised behaviour as the behaviour sanctioned by the society they live in and and every thing opposed to it as uncivilized. An interesting example is provided about the changing view of taking bath in western civilization. In nineteenth century, the Indian practice of taking bath every day was considered queer in western countries, where cold climate made it rather difficult to have bath too frequently. At that time Indians looked upon Westerners as filthy people. But subsequently advance of technology and prosperity made daily bath a common practice in western world, while poverty induced by British rule forced many poor Indians live in squalid conditions, which prompted Britishers to label Indians as people who did not care much about cleanliness and hygiene.
Anyhow we can say that from the seventeenth century onwards the western countries have in general made faster progress in technology than the eastern world, and even today the balance is tilted in favour of western countries.
In response to the earlier answer equating military power and victory with better civilization, I would like to point out that in history is full of example of less civilized people defeating more civilized people - for example victories of Attila, Hannibal, and Genghis Khan.
I think that the Europeans had evolved beyond the native peoples that they conquered. The Europeans had developed over centuries of time and had already had technology that the native people could only imagine. They also had an organized system of government, military armament, and a common language. All of these factors gave them more power over the native people. The other thing that the native people did not understand was just how many Europeans there were on the other side of the ocean. They just kept coming and coming.
I do not think that the native people did not have their own civilization. Their way of life worked very comfortably for them at the stage of their existence and the location until the Europeans came along. The may have eventually evolved to a different level if the need arose in time, but with the entrance of the Europeans there was a big shift that affected their ability to survive as a culture.