First of all I say that the girl is right in the way she has reacted. I even appreciate her for that.
About you and the situation:
Let me asses and analyse the given situation from two angles. One ,the general angle and two the specific case angle.
If I view it from a general angle, I can say that it is not a welcome attitude from a professor or teacher to have such attitude to a student under him. There is dominance –vulnerability situation. It may be just infatuation or a subjugation to a dominant and power yielding person by a helpless or vulnerable person. Here you are the dominant personality and the girl is the vulnerable helpless person. By normal ethics and moral standard your side is to blame. It can be termed as exploitation also. In that sense I would suggest you better terminate the peculiar relationship you want to have with her and treat her also as a student by a teacher.
Now, the second angle. You are an eligible bachelor. The girl (apart from the fact that she is your student) is of eligible age to marry. She has a right to choose you as her life partner if she finds you suitable for that. In that case she should feel convinced that yours is not a fleeting affection originated from outward attractions and opportunity of proximity and interaction. There is nothing wrong for an eligible bachelor to think of proposing to an eligible woman of marriageable age. But for that first you have to know her mind.
As a student, her priority is studies and completing the academic course successfully. She would have thought marriage as second priority and even if she had some favourable attitude to you, she would have thought that marriage can wait. She really wanted maturity and discretion from your side. But as the matter became public debated topic, she is really embarrassed. She would have naturally got shock and even doubted your maturity and ability to handle such matters.
Now, there is a side of taking care of her honour and dignity. Please do not harass and embarrass her by pestering or reminding and following up. Behave as a normal teacher and student.
In case you are serious of marrying her, with dignity and honour meet her elders (parents) without making it public. Keep her privacy intact. Diplomatically and discreetly open the topic (even directly) to her parents. Convince them that you are serious about this and would precede only if the girl also agrees for it. Assure them that her studies will not be discontinued and marriage can wait after the course or she can continue the studies even after marriage. If they also agree, proceed in the proper way how arranged marriages proceed.
Once matters decided you both can declare the matter to public. Then everything will be alright.
If the answer is negative from her or her parents side, accept it with grace and erase all the previous thoughts and expectations and be a good teacher taking things as a dream and behave normally. Do no make anything public. Avoid embarrassing her. You will have a good life ahead.
But before all this ask yourself whether you are serious of the matter and whether she is comfortable with this.