That would be true only if the US were handling this process extrajudicially. I'm still not sure that would be morally bad, per se, but it would certainly require the exercise of much more power and will.
But in this case, the process is being handled judicially. The people of the United Kingdom elected representatives, who through various means constructed the judicial system and entered into agreements with other countries, including the United States, concerning many things, including extradition. The process they set up to handle that is what is being followed today. This is why I don't really agree that the current situation is undemocratic. It may be that the UK's representatives didn't do exactly what the people would prefer in aggregate, but that is always a danger in representative democracy.
That's ignoring a lot of behind-the-scenes pressure tactics. In fact, your analysis would cast every one-sided deal or law that is the result of foreign pressure (or corruption, or lobbying) as "democratic", unless it's the result of full-on war.
Suppose I view my diet and exercise regimen as right - should I try to impose (not just suggest) it on others? A country can view both its own laws, and the sovereignty of other countries, as just.
Or look at it this way: Democratic countries tend to think highly of democracy. Higher than any single law resulting from democracy. By imposing your laws on the people of a different country, you are robbing them of that same democracy you so cherish. Would you give up democratic control of your government, if it resulted in passing a couple of laws you liked?
(For certain extreme examples, such as when that country is engaging in genocide or extreme human rights abuses, I would answer yes. But nothing in this case even approaches such severity.)