Downtown on the Riverwalk (not the touristy places):
Restaurant Le Reve - THE best restaurant in Texas, arguably in the U.S. Just amazing. Probably won't be able to get reservations at this point, but would be worth a shot. Expensive, but worth it.
Biga on the Banks -- An excellent restaurant and quite romantic. Outdoor dining on the river, if you wish.
A bit further out:
The Vineyards -- In a lovely setting of the Texas Hill Country, just outside San Antonio on Natural Bridge Caverns Road (3009). One of the most romantic settings you'll find. The dining room is glass on 3 sides, overlooking the vineyard below. They even do weddings there. Food is fabulous. It should be -- my grandson is the executive chef!
Good luck and have fun!