Here are the interests of the characters:
Paige: medicine, getting into medical school
Eddie: cars, girls
What do they do together? First, he annoys her and gets rejected. She appears beautiful and challenging. Then, she starts liking him but denies it for some reason, after a bit of persistence and doing some schoolwork together. They meet her family. They work at the same restaurant and she teaches him to slice meet. She also teaches him to do laundry. That's about it. Throw in flirting and you have the basis of their relationship.
Of course, once they kiss, everything changes. Now they touch all the time and smile a lot, and this distracts them from doing anything else. However, this is too boring to last long in a movie, even though it's supposed to be the good part (and shouldn't the good part of a relationship be interesting?). So they quickly get caught making out by the media, and then she finds out he's a prince and never told her. And then she dumps him for "lying" to her (by omission). How quickly "happily ever after" isn't!
As usual, she soon realizes her mistake and regrets it. If only she'd ever seen a movie like this one, she'd have realized it's better to think first and breakup second. So she goes back to him.
Then he's going to be King soon, so proposes marriage. She agrees without asking any questions. Isn't that strange? Why does becoming King mean he needs a wife immediately? Is it really a good idea to get engaged a couple days after a breakup? And what responsibilities does a queen have? And where will they live? And what will happen to her life and future plans? Will she still go to medical school? At the school she intended, or one in Denmark?
But considering stuff like that is a matter of reason. This is a movie about love. So she just agrees and finds out what it means later. She has a very busy schedule, most of which seems boring. Worse, he's busier and has to leave her in the middle of activities (which means sexual touching getting interrupted, because that's all they really do). This leaves her waiting around, alone.
So she dumps him again to back to the USA and go to medical school. She doesn't want to give up her career. It never occurs to her to try to get both things she wants. She just picks one and painfully gives up the other.
If he cared about her at all, he would have seen this problem coming in advance. I sure did, and I'm just an audience member. He could have been figuring stuff out like whether there is a medical school in Denmark she could go to (of course there is). She could have thought of this too it's not that hard.
So after she dumps him and leaves, he follows her and says "I'll wait." She can go to medical school, and whatever else she wants, and they'll be together afterwards. They still don't seem to realize she could go to school in his country and they don't have to be far apart. Regardless, she agrees, and the movie closes without further discussion: they will live happily ever after. The last two times that appeared to be true, and then she dumped him, were just bad luck, but there is no possible way they'll be unlucky again. Why not? Because they are in love. Being in love prevents bad luck, except when it doesn't. Umm. Yeah I give up, they are just dumb. Relationships should contains mechanisms to stabilize them against bad luck, and ameliorate the effects of bad luck. And not just bad luck (it wasn't actually a matter of luck), but problems of all sorts, whatever their cause. What sort of mechanism would work well? For a start, they should think about how they can get the things they want and solve problems, instead of resorting immediately to breaking up. And they should get to know each other over time and avoid any commitments or obligations which don't have specific and valuable function instead of just jumping into things and having faith that love will see them through.
Everyone knows that communication is the key to a good relationship. Or so they say. Yet movies like this are the norm, and movies featuring good communication are nonexistent. Romance novels, which have plenty of space for dialog, also feature poor communication. The truth is, reasoned discussion isn't sexy. Love doesn't like being talked about, and people enjoy it more when they aren't talking, they are just touching and smiling.
The best part of the movie was when they first approach the royal palace together:
Paige: Oh no.
Paige: You didn't tell me you lived with your parents.