Egads, this is hard! I'm going to have to say that I love the proposal scene in Kamisama Hajimemashite. I think this is probably the most romantic scene I've come across in any of the Anime I have watched (which I realized this morning, don't have many really romantic scenes, ship scenes, yes, but not really romantic). So, this is the one I've chosen!
In this scene, Tomoe finally realizes he loves Nanami. He has watched a lot of what her life was like before coming to the shrine in her memories and has fallen deeper in love with her by doing so and really learning more about her. In her memories, her younger self becomes sad because there is no one else at her home, she is all alone. He comforts her by telling her she'll have a rowdy household to look after later. She adamantly says she will never get married because she cannot rely on men. Tomoe gets caught up in her tirade and says that she is a poor judge of character, that HE could make her happy.
Nanami then throws the curve ball of 'I won't get dragged into a loveless marriage. Do you love me?'. Tomoe seems like he is surprised and gives it thought, and then tells her that he DOES love her, and asks her to marry him. I was wondering whether he was just humoring her when I first watched this. But then they show that he kinda got carried away with his feelings and went all "wait, what did I just do?". Which was really neat to me, that they finally got cold, aloof Tomoe to let down his guard and really say what he feels. And that they let the audience finally know how each character really feels about the other!
Overall this scene was sweet and light and gave us a chance to see the romantic side of Tomoe (which was what was left, given we've known Nanami's feelings for a while).
Anyways, A long explanation, but this was my favorite (and potentially the only(?)) really romantic scene I've seen in any anime!