The most romantic place I've been is Santorini, Greece. The magic of that island can't be expressed by words, and truly needs to be experienced in order to be understood. These pictures don't do the beauty justice; I highly recommend carving out time to visit this magical world.
Getting there is going to be difficult - not in terms of cost or miles, necessarily, but in terms of time. From either the east or west coast, you're likely going to make two stops and take 20+ hours of travel time. I would highly recommend saving up United miles for the journey: 140k miles round-trip in business class is a small price to pay, especially if you're able to score a Swiss Air seat. Cut that amount in half if you're flying coach.
My last note on this: stay in Oia, nowhere else. Hotel points don't help on Santorini, because almost all the hotels are boutique - and Oia at sunset will be the peak of your travels. Credit cards with travel statement credits could do wonders to bring the cost of your vacation down. Also, consider off-season if price is a concern - the island is less beach-oriented so hot weather isn't necessary.