My take:
As some of you know, I'm a former military servicemember. Am I a 'veteran'? Well, I don't know that I deserve that moniker, but I did serve my country.
I often take shots at the military, because I'm intimately familiar with a lot of the BS that servicemembers are subjected to by civilian authorities.
That said, regardless of any actions they might take, I expect the White House to hold our nation's military in highest esteem and for Biden to come out essentially mocking McCain for being a 'good soldier' is a smack in the face to every good American, because in so doing he mocks the very important job our military does for us all.
I'm perfectly fine with John Q. Citizen making a remark like that, but it's entirely inappropriate for someone who aspires to the White House.