No, because the term refers to the time when you reach young adulthood, not when you're born. This is also why many of the people who like to "slam Millennials" are so funny: They're Millennials themselves.
Now, being the wild card that I am, I would like to propose a crazy counter-argument to the "Millennials don't watch older movies" debate. Here it is: People who love movies, watch older movies; People who don't, not so much. Across all generations.
If anything, common sense dictates that Millennials would watch more films than previous generations, both because they live in a time when movies are more accessible to watch than ever (we now have streaming and fast internet, while older generations had to rely on the movie selection at their local video store, if they had one at all), and because Media Research shows a huge increase of media content consumption over the years.
"Millennials don't watch older movies" is just your typical Old Man Argument, that's been used since the dawn of time by older generations to "criticize" younger generations. There's no real substance behind it. Jesus, these kids today...