Can primary care physian be held responsible for not treating patient with pneumonia correctly, resulting in ARDS, ending in death 3 weeks later?

The answer here is a very qualified yes, depending upon the circumstances of the case, and of course, how long ago this occurred.  You need to consult a medical malpractice attorney, and perhaps more than one, to get a good sense of what you should do.  Most of these attorneys do not charge for an initial consultation, so inquire about that beforehand.  On the other hand, consult lawyers experienced in this kind of law--it is often complicated.