Romance does not have to be a part of the game, but it's nice to have the option. If your character is wandering around everywhere with a group of people s/he would in your opinion find attractive, the ability to pursue that attraction if you choose is a natural outgrowth of the story so makes sense. It's harder to implement in single character RPGs than in party based RPGs, but still not impossible.
Best implementations: Dragon Age implements the option superbly, giving you a great choice of romance options (Including 'none'). Probably the best party based version, though Jade Empire is also good and the classic PS:T is also. For single character games, only The Witcher implements romance well, AFAIK, giving you both options and a good basis for those options in that the possible romances arise as part of the story and the characters are recurring and strongly involved in the story itself, so Geralt knows them pretty well.
I'm not so keen on romance being an absolute requirement, as it takes choice away from the player in a very key area which makes it pretty difficult to define that one key aspect of your character. JRPGs seem to use this option most, but they are more like interactive films than RPGs anyway and I play them on that basis. I don't really expect to be making important character choices in those.
That said, romance is pretty much pointless in an ARPG, where the only point of the game is to knock up an enormous kill count. I can't see the Diablo games being at all improved by the addition of romance options - in fact, it would probably be nothing but an annoying distraction for most Diablo players. Such games are not about playing a character, after all. I love them when I'm in the right mood, don't get me wrong, but they still are not RPGs.
