No it isn't a payer.

But the larger context is that the 1st amendment doesn't specifically call out prayer but instead prohibits establishment of a national religion.

As such, the government isn't permitted to endorse religious belief. "One nation, under god" isn't specific I guess, but it is tied to the belief in a god, as opposed to no belief or belief in multiple gods. It sure seems like endorsing some religions over others or none.

I fail to see how this is still allowed in school except that nobody is required to say the pledge. (As far as I know).