Nope, you've been lied to. Stories of "Vets being spat upon" is exactly that. Stories.
They didn't start showing up until the 1980's, when Conservatives started inventing various versions of a "Stabbed in the Back" myth to explain Nixon's surrender in Vietnam.
Vietnam Vets were treated horribly, BY THE GOVERNMENT, and THE BIG CORPORATIONS THAT PROFITED from the war.
I should point out that my parents were pretty conservative (mostly) Catholics and up until about 2008 or so, I voted Republican.
But almost no one argues today that the Vietnam War was a good idea. Because it wasn't. It was sold on lies, and perpetuated because neither side wanted to be the one holding the bag when it did go south.
So instead, they shifted to "The Vets were done dirty" by the hippies (When often they were of the same age group and often the same people.)