Is it correct to say?
He rushed to her.

(A man and a woman were having dinner at a restaurant.Then the woman left. Suddenly, the man realized that something bad would happen to the woman. He immediately ran after to warn her about danger.)

He rushed to her... is fine. Also, he rushed over to her, or he rushed after her. I like "he rushed after her" ... if he is hurrying after her out of the restaurant.