We hate to bring this news to you, but there is no "great silent majority" of American society. Nixon isn't really speaking to anybody in particular. But at the same time, he is speaking to everybody.

We know—mind blown, right?

Nixon uses the phrase "great silent majority" as a double-edged sword. On one side, he is trying to get his constituency all riled up and is irritated that faith in the United States was starting to slip. On the other side, he wants to use this phrase to condemn and insult the anti-war and hippie crowds.

Look at what follows his "great silent majority" line:

And so tonight—to you, the great silent majority of my fellow Americans—I ask for your support.

[…]

The more support I can have from the American people, the sooner that pledge can be redeemed; for the more divided we are at home, the less likely, the enemy is to negotiate at Paris.

[...] North Vietnam cannot defeat or humiliate the United States. Only Americans can do that. (131.1-134.3)

Nixon is trying to make it seem like he wants the nation to come together as a unit. The thing is, though, when people do come together, they come from all over the place and have different beliefs and perspectives. One of those beliefs might have been that the United States should fully withdraw from Vietnam.

Nixon doesn't want anybody like that coming together into his group, though. In fact, he wants to blame people like that for the enemy not showing up to the negotiating table. So, even though his use of "great silent majority" may come across as a great quote for unification and working together, it's actually a tool to criticize certain people while putting others up on a pedestal.