She can take you to court, but all you both have is, from your description, an unwitnessed verbal agreement. Unless there was someone else there who witnessed your saying that, she will have no proof.
On the other hand, it isn't ethical for you to back out on your word if you took responsibility for ensuring that the dog would get a good home, either.
Perhaps you can resolve it by finding a proper rescue that will help rehome the dog without it being euthanized because neither of you are appropriate. Look on Petfinder.com in your area, and find her a good rescue group to take the dog into foster, if that's the best you can do. If it a particular breed, there is also breed rescue to approach - but do make an effort to do better than telling her just to dump the dog at the local shelter.
By going to court, you will both be wasting valuable time of municipal employees that have far more important things to resolve, and it won't help the dog anyway. Why not buck up and help her find the right place for the dog, if it's not with you.