Not quite. It turns out, from fascinating polling data on the Vietnam war, that the opposite is true.
There were many polls on public opinion during the war, and they show a consistent pattern by age. Young people were more likely to support the war at the beginning, when it was popular, and more likely to support it at the end, when it was not.Comparison to polling data about Iraq is left as an exercise for the reader. (Hint: Americans don't like either war.)
Other common beliefs about public opinion on the Vietnam war are also false. Educated people were more likely to support the war, not less. There is not as much data on the subject, but draft status did not seem to affect opinions on the war.
Some common beliefs about the war are correct. Women were more dovish than men, and blacks more dovish than whites. All the patterns that I have mentioned were also found in public opinion during the Korean War and World War II.