Grammatically, it's correct. It doesn't sound like something natural when having a conversation with someone.

I think it would be more natural to say 'He is quite heedless of his son.'

Yes, it is grammatically correct. Here in the United States, I don't hear the word "heedless" very often.

Hi Furqi. I believe that it is a gramatically correct sentence. However, in the U.S., few people use the word "heedless." "Heed" is the only semi-common form used (to mean "pay attention to"). Some people would have to think about that sentence to understand what it means. We'd use another word (each of which doesn't have the same exact meaning): unthoughtful, unmindful, indifferent, uncaring, "... unconcerned about his son," and so on.