That is the problem, yes.
No you cannot just buy that diaphragm from the dealer. As far as I know, nobody sells just the diaphragm separately. You will need to purchase a new petcock assembly.
Yes, that little rip will stop the fuel from flowing to your engine, and cause a lot of drive-ability problems. The leaking fuel from the downward pointing nipple was a clear indication of the torn diaphragm.