Is it possible to go cross country, Atlantic to Pacific ocean, just using metro public transportation?

Can't use amtrack or greyhound. An example would be NYC Subway, NJ Path, NJ Transit and BART

Not if the country is the USA. Can't see it using metro only. Regional bus companies, maybe. Caravan for Commuters (vanpool) might make up some of the in-between areas.

Not using metro public transportation, our cities are too far apart
However you can use public transportation on Amtrak and get coast to coast, my parents did it (it costs just as much or more than flying, and takes much longer, but they said it was very scenic)

I seriously doubt you could do this. You must live in a metro w/decent transit to even ponder it (NYC, I'm guessing?) but think of all the miles and miles of rural counties spread across the midwest and western states.

Starting in NY, you could probably make it as far as Philadelphia, maybe Pittsburgh. Maybe. But from there on out, it would be hard going is my guess.

No it is not, metro public transportation only opperates in the metropolitan city areas and not in the rural areas in the united states. There are hundreds of miles in between the larger cities that have metro where there is no public transportation. The only way would be greyhound but it takes several days and is a misserable ride. I have made this trip on greyhound and it sucks.