If it was a budded rose bush and was broken of at the root it will not come back as the bush you purchased. The root stock may grow but that would probably be Dr. Huey, a once blooming climber with small red blooms.

I doubt the top will root in water, I had an own root bush that broke off like yours. I cut it back some, dipped it in rooting hormone, placed it in a pot with potting soil and covered it with a large jar. That was 6 weeks ago and it appears to have rooted.

What kind of rose is it?

I am not shore of the type I bought two rose bushes at wall mart this spring they have large blooms I thought they would be yellow however one is red the other is pink so even If I kept the name it may not be correct. the One thats bean trampled was red and in bloom.