You do not have to raze or reduce the city to 1 to hold it. But, depending on culture, you may have to remove the civ from the board to hold their cities.
As I took Aztec, the cities began flipping during the campaign. Their total culture was about equal to mine. When I captured a city, I had only the basic diamond of control, of course, and move troops out to the next conquest as soon as they healed, and resistance was down. Cities began flipping behind me. Not all, but one every turn or so. When they did, I noticed that their zone of control included everything not in my basic diamonds. Even thought the city now had no culture installed, they did have a massive culture memory, as mentioned in Dan's post.
There is some irony here. Democracies don't like war, in general. THe US had massive internal resistance to involvement in war in WW I, WW II, and Vietnam. My guess is that fully half the peopl did not want to be in those wars. Didn't bring down the government, but I suspect that is a possibility.
My democracy has fallen twice in this campaign. When it came back out the first time, I made sure every city was at least content, leaving no unhappy people anywhere except to avoid starvation. Maybe I should have let them starve, because one turn later it collapsed again.
True, I was the agressor, and it was not a just war. BAsic reason: to win the game, to rule the world.
Here is the irony> Winning by culture requires a high score, culture more than twice your nearest civ. At least one of the civs will build culture--in my current game it is (was) Aztec and China. Every cultural improvement possible in every city. If both of you do that, then the only way to beat his culture is to remove it. If both have 30 cities producing max culture, it is logically impossible to double his score. Reducing his empire size will slow his culture growth, but the old accumulated culture remains.
I have won by culture without war, at chieftain level. Not on higher levels... The only way I see to win by culture is to make war. ANd then you might as well just conquer them.