Let's break this down for some analysis and try and leave out the tribalism and America hating lunacy, though I confess these days I hate it more than some of you.
First some categories
Military/Conquest
Science/Invention
Culture
Commerce
Philosophy/Knowledge/Ideas
Religion
Military
1. Mongols (greatest warriors in history bar none)
2. America (nuclear weapons, aircraft carrier, nuclear missile sub, WWI,II,etc)
3. England (seapower, ruled an empire on which the sun never set)
4. Rome (roman legion, nuff said)
5. Arabia just edges Germany(not particularly fair as Arabia is a conglomerate 'civ' but Islam spread mainly through conquest over a large area plus the Germans got rolled both times and hitler was a flash in the pan)
Science
1. China (gunpowder,fireworks,crossbow,plow,cast iron, many many more)
2. Arabia (Arabia is sort of a conglomerate for islamic world but lets give them their props for science while Europe was in dark ages)
3. America (rank 3 is exceedingly stingy for the only nation to put a man on the moon, invented the bomb, internet (thanks Al), computers, the automobile, sequenced the genome, space shuttle, I would continue but it would grieve you America bashers too greatly I fear, plus a large number of the inventions are warfare based which grieves me a bit.
Not even going to continue with this category, after the first 3 no one else can come close, the renaissance pretty much has to be divided among several european countries and I suppose you could argue for Russia or Germany having one of the spots.
Culture
1. America (love it or hate it you know its true)
2. England (English is the world language of business for a reason)
3. Arabia (Arabia makes the list here as well as religion)
4. France (can't think of a good reason...but anyway)
5. Spain (sort of a stretch as they mostly only left a language behind)
Commerce(getting tired of typing, reasons should be self-evident)
1. England
2. America
3. China
4. Rome
5. Arabia
Philosophy/Ideas
1. Greece (philosophy, birthplace of democracy)
2. India
3. China
4. Arabia
Again you have to divvy up the renaissance too much among Europe and can't give it to one particular country.
Religion
1. India
2. China
3. Arabia
4. Rome (give them credit for Christianity)
5. America (for good or ill, modern protestant evangelism originates here)
So to sum up, it looks like we have
1. America
2. China
3. England
4. Arabia
5. Rome
While the near future may see china rise and america continue to decline, America narrowly beats out china atm. England edges Arabia, because again 'Arabia' is not a civ per se. And the perennial favorite Rome, squeezes in to number 5.
To those of you who would say that I am slanted towards Anglo-Saxon countries, I would say that is sour grapes. I personally don't consider the reasons countries are considered 'great' to be especially honorable or noble things. America may not have stood the test of time but it is a country like the world has never known and remains for now the 'greatest' nation to ever exist, that is just the cold hard facts.
I will admit to being biased against ancient nations however. I don't see how anyone can say Egypt should always be included, it is a dead culture. They worshipped the stars and left some monuments, now they are dust and nothing else remains of them. Same can be said for the Inca, Maya, Babylonians, etc. I do have a soft spot for the Persians who get rolled over and they always come back but hardly a reason to make the top 5 at 5.
Edit: I guess I could have read the poll first and saw that Arabia wasn't included, but hey Civ does so I am sticking by my rankings!