Is a son considered undutiful to his parents if he marries a girl whom his parents rejected for illogical and un-Shar'i reasons, despite his compatibility with her and his attachment to her, and after his parents cut ties with him and accused him of being undutiful?

Marriage via the internet is often fraught with risks, but if the girl is religious and of good character, you are not obliged to obey your parents in preventing you from marrying her, especially if you would be harmed by not marrying her. For obedience to parents is not obligatory in matters that harm the child. However, if you would not be harmed by leaving her, then it is better to leave her and search for someone else whom your parents approve of, for honoring one's parents is among the best acts of devotion. In any case, you must honor them and strive to gain their satisfaction.